




































































• 
















- 








































V 

































ELEMENTS 



OF THE 



DIFFERENTIAL AND INTEGRAL 



CALCULUS 



BY CHARLES DAVIES, LL. D., 

AUTHOR OF ARITHMETIC, ELEMENTARY ALGEBRA, ELEMENTARY GEOMETRY, 
ELEMENTS OF SURVEYING, ELEMENTS OF DESCRIPTIVE GEOME- 
TRY, ELEMENTS OF ANALYTICAL GEOMETRY, AND 
SHADES SHADOWS, AND PERSPECTIVE. 



IMPROVED EDITION. 



PHILADELPHIA: 

PUBLISHED BY A. S. BARNES & CO., 

21 Minor Street. 
1843. 



QA503 



Entered according to the Act of Congress, in the year one thousand 
eight hundred and thirty-six, by Charles Davies, in the Clerk's Office 
of the District Court of the United States, for the Southern District of New 
York. 



GIFT 

ESTATE OF 

WILLIAM C. RIVES 

APRIL, 1940 



C. SHERMAN, PRINTER, 

No. 19 st. Jan es Street, 

PHILADELPHIA. 



PREFACE. 



The Differential and Integral Calculus is justly con- 
sidered the most difficult branch of the pure Mathematics. 

The methods of investigation are, in general, not as 
obvious nor the connection between the reasoning and 
the results so clear and striking, as in Geometry, or in 
the elementary branches of analysis. 

It has been the intention, however, to render the sub- 
ject as plain as the nature of it would admit, but still, 
it cannot be mastered without patient and severe study. 

This work is what its title imports, an Elementary 
Treatise on the Differential and Integral Calculus. It 
might have been much enlarged, but being intended for 
a text-book, it was not thought best to extend it beyond 
its present limits. 



4 PREFACE. 

The works of Boucharlat and Lacroix have been 
freely used, although the general method of arranging 
the subjects is quite different from that adopted by 
either of those distinguished authors. 

The present is a corrected, and it is hoped an improved 
edition. The first chapter has been entirely re-written, and 
some of the other parts of the work have been considerably 
altered. 

West Point, March, 1843. 



CONTENTS 



CHAPTER I. 

Constants and variables, .... 

Functions defined, ...... 

Increasing and decreasing functions, 

Implicit and explicit functions, 

Differential coefficient defined, 

Differential coefficient independent of increment, 

Differential Calculus defined, . 

Equal functions have equal differentials, 

Reverse not true, ..... 



Page. 

9 
9 
10 
11 
16 
20 
22 
23 
23 



CHAPTER II. 

Algebraic functions defined, .... 25 

Differential of a function composed of several terms, . 26 

" " the product of two functions, . . 27 

" " " any number of functions, 28 

" " a fraction, 29 

Decreasing function and its differential coefficient, . 30 

Differential of any power of a function, ... 30 

" of a radical of the second degree, . . 31 

" coefficient of a function of a function, . 33 

Examples in the differentiation of algebraic functions, 34 

Successive differentials — second differential coefficient, 39 

Taylor's Theorem, 43 

Differential coefficient of the sum of two variables, 43 

Development of the function u — (a -f- x) n , . . 46 

" second state of any function, . 47 

Sign of the limit of a series, ..... 47 



6 CONTENTS. 

Page. 

Cases to which Taylor's Theorem does not apply, . 48 

Maclaurin's Theorem, ...... 50 

Cases to which Maclaurin's Theorem does not apply, 53 

Examples in the development of algebraic functions, . 54 

CHAPTER III. 

Transcendental functions — logarithmic and circular, . 55 

Differential of the function u = (f, . . . 55 

" " logarithm of a quantity, ... 58 

Logarithmic series, ...... 59 

Examples in the differentiation of logarithmic functions, 62 

Differentials of complicated exponential functions, . 64 

" " circular functions, .... 66 

" " the arc in terms of its functions, . 70 

Developmentof the functions of the arc in terms of the arc, 73 

Development of the arc in terms of its functions, . 75 

CHAPTER IV. 

Partial differentials and partial differential coefficients 

defined, ........ 79 

Development of any function of two variables, . . 80 
Differential of a function of two or more variables, . 82 
Examples in the differentiation of functions of two va- 
riables, ........ 85 

Successive differentials of a function of two variables, 86 

Differentials of implicit functions, .... 89 

Differential equations of curves, .... 93 

Manner of freeing an equation of constants, . . 96 
" " the terms of an equation from ex- 
ponents, ....... 97 

Vanishing fractions, ....... 98 

CHAPTER V. 

Maxima and minima defined, 105 

General rule for maxima and minima, . . . .108 

Examples in maxima and minima, .... 109 

Rule for finding second differential coefficients, . . 112 



CONTENTS. 



CHAPTER VI. 

Page. 

Expressions for tangents and normals, . . . 116 

Equations of tangents and normals, . . . .118 

Asymptotes of curves, ...... 122 

Differential of an arc, . . . . . .125 

" " the area of a segment, . . . 127 

Signification of the differential coefficients, . . 128 

Singular points defined, . . . . . . 132 

Point of inflexion, . . . . . . .133 

Discussion of the equation y = bzk c(x — a) m , . 134 
Condition for maximum and minimum not given by Tay- 
lor's Theorem — Cusp's, 139 

Multiple point, 143 

Conjugate or isolated point, ..... 144 

CHAPTER VII. 

Conditions which determine the tendency of curves to 

coincide, 147 

Osculatrix defined, 150 

Osculatrix of an even order intersected by the curve, . 152 

Differential formula for the radius of curvature, . 154 

Variation of the curvature at different points, . . 155 

Radius of curvature for lines of the second order, . 156 
Involute and e volute curves defined, . . . .158 

Normal to the involute is tangent to the evolute, . 160 
Difference between two radii of curvature equal to the 

intercepted arc of the evolute, . . . 162 
Equation of the evolute, . . . . . .163 

Evolute of the common parabola, . . . . 164 



CHAPTER VIII. 

Transcendental curves defined — Logarithmic curve, . 166 

The cycloid, 169 

Expressions for the tangent, normal, &c, to the cycloid, 171 

Evolute of the cycloid, 173 

Spirals defined, ....... 175 



CONTENTS. 



INTEGRAL CALCULUS. 

Page- 
Integral calculus defined, 189 

Integration of monomials, . . . . . 190 

Integral of the product of a differential by a constant, 192 

Arbitrary constant, . . . . . . 194 

Integration — when a logarithm, . . . . .194 

Integration of particular binomials, . . . 195 

Integration — when a logarithm, . . . . .195 

Integral of the differential of an arc in terms of its 

sine, . . . . . . . . .196 

Integral of the differential of an arc in terms of its 

cosine, . . . . . . . .198 

Integral of the differential of an arc in terms of its 

tangent, . . . . . . . .199 

Integral of the differential of an arc in terms of its 

versed-sine, 200 

Integration by series, ...... 201 

" of differential binomials, .... 207 

Formula for diminishing the exponent of the parenthesis, 212 
Formulas for diminishing exponents when negative, . 213 
Particular formula for integrating the expression 
dx q 



214 



V2ax— x 2 
Integration of rational fractions when the roots of the 

denominator are real and equal, . . . 216 
Integration of rational fractions when the roots are 

equal, 221 

Integration of rational fractions when the denominator 

contains imaginary factors, .... 226 

Integration of irrational fractions, .... 233 

Rectification of plane curves, .... 243 

Quadrature of curves, 250 

" of curved surfaces 261 

Cubature of solids, 269 

Double integrals, ...... 274 



DIFFERENTIAL CALCULUS. 



CHAPTER I. 
Definitions and Introductory Remarks. 

1. All the quantities which are considered in the Dif- 
ferential Calculus may be divided into two principal 
classes : constants and variables. Each constant retains 
the same value throughout the same investigation ; but 
the variable quantities are subjected to certain laws of 
change, in consequence of which they may assume in suc- 
cession, an infinite number of different values, without 
changing the form of the expression into which they enter. 

The constant quantities are generally designated by the 
first letters of the alphabet, a, 6, c, &c. ; and the variable 
quantities by the final letters, oo, y, z, &c. 

2. If two variable quantities are so connected together 
that any change in the value of the one necessarily pro- 
duces a change in the value of the other, they are said to 
be functions of each other. 

Thus, in the expression 

y = ax, 



10 ELEMENTS OF THE 

y and x are functions of each other ; for, if any change 
be made in the value of x, a corresponding change will 
take place in that of y ; and leciprocally. 

3. When the value of one variable depends on that of 
another, as in the expression 

y — ax, or y — ex 2 , 

if we attribute at pleasure any increment to one of the 
variables, a corresponding change will take place in the 
other ; and hence, if one of them be supposed to increase 
or decrease according to any independent or arbitrary 
law, a corresponding change of the other will take place 
according to the law of relation which exists between 
them. The one to which the arbitrary increment is 
given, is called the independent variable, or simply the 
variable, and the other is called the function. 

4. The relation between a function and its variable is 
generally expressed thus : 

V =/(*). 

in which / is a mere symbol, denoting that y and x are 
functions of each other. The expression is read, y a 
function of x, or y equal to a function of x. 

The mutual dependence of one variable on another 
may also be expressed under the form 
f(x,y) = 

in which y is a function of x, and x a function of y. 

5. Functions are either increasing or decreasing. An 
increasing function is that which increases -when its vari- 
able increases, and decreases when its variable decreases. 



DIFFERENTIAL CALCULUS- 11 

Thus, in the expressions 

y = ax 2 , u = (a + a?) 2 , 
y and u are increasing functions of x ; since, if x be in- 
creased,?/ and u will both increase ; and if x be diminish- 
ed y and u will both decrease. 

A decreasing function is that which increases when its 
variable decreases, and decreases when its variable in- 
creases. Thus, in the expression 

1 

X 

y is a decreasing function of x ; for, if x is decreased 
y will increase ; and reciprocally. 
In the expression 

y = {a- x) 2 , 

y will decrease while x increases between the limits of 
zero and a ; but will increase with x for all values of x 
greater than a. Hence, y is a decreasing function of x 
for all values of x less than a, and an increasing func- 
tion of x for all values of x greater than a. 

6. Functions are either explicit or implicit. An ex- 
plicit function is when the value of the function is di- 
rectly expressed in terms of the variable on which it de- 
pends. Thus, in the expressions 

u = bx 2 , y — Va 2 — x 2 , 
u and y are explicit functions of x. 

An implicit function is one where the value is not 
directly expressed in terms of the variable. Thus, in 
the expressions 



12 ELEMENTS OF THE 

au 2 + ex 2 = bx 2 , y 2 + x = a 2 — a? 2 , 
w and y are implicit functions of x. These expressions 
may be written under the form 

f(u,x) = 0, and /(*/,#) = 0. 
The relation between an implicit function and its variable 
may also be expressed by means of two or more equa- 
tions. Thus, if we have 

z = ay 2 , and y = ax, 

z is an implicit function of x. 

These expressions may be written 

? =f(y\ and y =f(x) ; 
or we may write 

f(z,y) = 0, and /(y,ff)=0. 

7. Functions are either algebraic or transcendental. 

8. An algebraic function is one in which the relation 
between it and its variable can be expressed algebraically 
— that is, by addition, subtraction, multiplication, division, 
or the extraction of roots indicated by constant indices. 
Thus, in the expressions 



u — ax 2 + ex, y = -y/a 2 — x 2 , 
u and y are algebraic functions of x. 

9. Transcendental functions are those in which the 
relation between the function and its variable cannot be 
determined by methods purely algebraic. For example : 

u = a x , u= log. x, u = sin x, 
are transcendental functions. 



DIFFERENTIAL CALCULUS. 13 

When the variable enters as an exponent, it is called 
an exponential function ; when it enters as a logarithm, 
it is called a logarithmic function ; and when it enters as 
a sin., tang., cos., &c, it is called a circular function. 
Thus, in 

u = a x , u is an exponential function of x ; 

u = logo?, u is a logarithmic function of x ; 

u = sin x, u is a circular function of x. 

10. Although the values of the function and variable 
may be changed at pleasure without affecting the values 
of the constants with which they are connected, there is, 
nevertheless, a relation between them and the constants 
which it is important to consider. 

If, in the equation 

a particular value be attributed either to x or'y, the other 
will be expressed in terms of this value and the constant 
quantities which enter into the primitive equation. Thus, 
in the equation 

y — ax + b, 

if a particular value be attributed to a?, the corresponding 
value of y will depend on the value assigned to x, and on 
a and b ; or if a particular value be attributed to y, the 
corresponding value of x will depend on that value, and 
on a and b. The same will evidently be the case in the 
equation 

x* + y* = R 2 , 



14 ELEMENTS OF THE 

or in any equation of the form 

y =*/<». 

Hence, we see that, although the changes which take 
place in the values of the function and variable are entire- 
ly independent of the constants with which they are con- 
nected, yet their absolute values are dependent on those 
constants. 

11. Since the relations between the variables and con- 
stants are not affected by the changes of value which the 
variables may experience, it follows that, if the constants 
be determined for particular values of the variables, they 
will be known for air others. 

Thus, in the equation 

X 1 + yl = R*, 

if we make #=0, we have 

y = ±R; 
or, if we make y=0, we have 

x=dbR. 

12. The function y, and the variable x, may be so re- 
lated to each other as to reduce to at the same time. 
Thus, in the equation 

y 2 = 2poc, 

which may be placed under the general form 

f(x,y) = 0, or y=f{oo), 

if we make x = 0, we have y = 0, or if we make y = 0, 
we shall have x = 0. 



DIFFERENTIAL CALCULUS. 15 

13. We have thus far supposed the function to depend 
on a single variable ; it may however depend on several. 
Let us suppose, for example, that u depends for its value 
on x, y, and z ; we express this dependence by 

If we make x—0, we have 

u =f(y> z ); 

if we also make y=0, we have 

u = f{z); 

and if, in addition, we make z = 0, we have 
u= a constant, 

which constant may itself be equal to 0. 
If the function be of the form 

u = b + ax -f- yx 2 + zx 2 , 

in which one of the variables is a factor of several of the 
terms, then, if x = 0, we shall have 

u =6; 

or, if x were a factor of all the terms, we should have, 

for x = 0, u = 0. 

14. Let us now examine the change which takes place 
in the function, in consequence of any change that may 
be made in the value of the variable on which it depends. 

Let us take, as a first example, 

u = ax 2 , (l) 
and then suppose x to be increased by any quantity h. 



16 ELEMENTS OF THE 

Designate by u' the new value which u assumes, under 
this supposition, and we shall have 

u' = a(x-\- h) 2 , 
or, by developing, 

u' = ax 2 + 2axh + ah 2 . 
If we subtract the first equation from the last, we shall 

have 

vf — u = 2axh -f- o^ 2 ; 
hence, if the variable x be increased by h, the function 
will be increased by 2axh + ah 2 . 

If both members of the last equation be divided by h 9 
we shall have 

= 2ax + o^j 



which expresses the ratio of the increment of the variable 
to that of the function. 

Let us take, as a second example, 
u = x 3 , (2) 
and suppose x to be increased by a quantity h; desig- 
nating by v! the new value which u assumes under this 
supposition, and we shall have 

u' = (x + Kf, 
and, by developing, 

uf = a? + 3x 2£ + 3^2 + # 

By transposing a: 3 , substituting for it its value w, and 
then dividing by h, we have 

0—l = 3x 2 + 3xh+h a . 



DIFFERENTIAL CALCULUS. 17 

From equation ( 1 ) we have 

= 2ax -f- ah ; 



h 

and from equation (2), 
vf — u 



3x* + 3xh + h 2 . 



h 

15. Let us now observe that the numerator in the first 
member of each of the above equations, is the difference 
between the primitive function u, and the new value vf , 
which arose from giving an increment h to the variable 
x, of which wis a function. Hence we see, that the first 
member of each equation is equal to the increment of 
the function divided by the corresponding increment of 
the variable. 

If we examine the second members of these equations, 
we find a term in each which does not contain the in- 
crement h, viz. : in the first, the term 2ax, and in the 
second, Sx 2 . If now, we suppose h to diminish, it is 
evident that the terms 2ax, and Sx 2 , which do not contain 
h, will remain unchanged, while all the terms which 
contain h will diminish. Hence, the ratio 
vf — u 

in either equation, will change with h, so long as h re- 
mains in the second member of the equation ; but of all 
the ratios which can subsist between 
vf — u 

is there one which does not depend on the value of h ? 
We have seen that as h diminishes, the ratio in the first 

2 



18 ELEMENTS OF THE 

equation approaches to 2ax, and in the second to 3x 2 ; 
hence, 2ax and 3a; 2 , are the limits towards which the 
ratios approach in proportion as h is diminished ; and 
hence, each expresses that particular ratio which is in- 
dependent of the value of h. This ratio is called the 
limiting ratio of the increment of the variable to the 
corresponding increment of the function. 

16. We are now to explain the notation by means of 
which this limiting ratio is to be expressed. For this 
purpose let us resume the equation 

u — u _ ., 
= 2ax 4- an, 

h 

and represent by dx the last value of h, that is, the value 

of h, which cannot be diminished, according to the law of 
change to which h or x is subjected, without becoming ; 
and let us also represent by du the corresponding dif- 
ference between w'and u; we then have 

dx 
The letter d is used merely as a characteristic, and the 

expressions du, dx, are read, differential of v., differential 
of x. 

It may be difficult to understand why the value which 
h assumes in passing to the limiting ratio, is represented 
by dx in the first member, and made equal to in the 
second. We have represented by dx the last value of h, 
and this value forms no appreciable part of h or x. For, 
if it did, it might be diminished without becoming 0, and 
therefore would not be the last value of h. By designa- 
ting this last value by dx, we preserve a trace of the 



DIFFERENTIAL CALCULUS. 19 

letter x, and express at the same time the last change 
which takes place in h, as it becomes equal to 0. For a 
like reason the last difference between uf and u is desig- 
nated by du. 

The limiting ratio in equation (2) is 

^ = 3*3. 
ax 

The limiting ratio of the increment of the variable to 
that of the function, which has been found in the pre- 
ceding equations, is called the differential coefficient of\x 
regarded as a function of x. 

17. Let us take, as another example, the function 

u = ax^ : (3) 
if we give to x an increment h, we shall have 

u' — ax^ + 4:axVi -f- 6ax 2 h 2 -f- 4axh 3 + «/i 4 , and 

JL-ZJf = 4ax 3 + 6ax 2 h + 4ah 2 -f ah 3 , 
n 

and, by taking the limiting ratio, we have for the differ- 
ential coefficient, 

dx 

18. If it were required to find the differential of the 
function u, after we had formed its differential coefficient, 
it could be done by simply muliplying the differential 
coefficient by the differential of the variable ; thus, from 
equation (l) we should have 

du = 2axdx ; 
from the second, 

du = Sx 2 dx ; 



20 ELEMENTS OF THE 

and from the third, 

du = Aax 3 dx. 

The differential of each function may also be written 

under the following form: 



Eq. 1, 


-j-dx = 2axdx ; 


Eq. 2, 


du 

-ydx = 3x 2 dx ; 

dx 


Eq. 3, 


du 

-j-dx = 4:ax 3 dx; 

dx 



which, indeed, is nothing more than finding the differen- 
tial of the function by multiplying the differential coefficient 
expressed in the first member of the equation, by the dif- 
ferential of the variable. 

19. Let us now examine each of the three equations 
which we have considered, and observe the form of the 
expression for the difference between the two states of 
the function u. 

From the first equation we have 

u' — u — 2axh -f- ah 2 ; 
from the second, 

tf _ u = Sx 2 h + Sxh 2 -f h 3 ; 
and from the third, 

u' -u= 4ax 3 h + 6ax 2 h 2 + Aaxh 3 + ah\ 

We see in each of the expressions for the difference 
between the two states of the function u, that the first 
term of the difference contains the first power of the in- 
crement h, and that the coefficient of this term is the 
differential coefficient of the function u, or the limiting 



DIFFERENTIAL CALCULUS- 21 

ratio of the increment of the function to that of the vari- 
able. This differential coefficient is, in general, a func- 
tion of x. 

If, now, in either of the expressions, we represent the 
differential coefficient, or limiting ratio, by P, and all the 
following terms of the difference by P'W (in which P / 
will in general be a function of h), the difference may be 
written under the form 

u' -u^Ph + P'h 2 ; 
and we shall assume that what has been proved in regard 
to the three forms of the function u which we have con- 
sidered, is equally true for all other forms. This form, 
for the difference between the two states of the function, 
is important, and should be carefully remembered. If, 
then, we have a function of the form 

u =f(x), 
and give to x an increment h, we shall have 
vf - u = Ph + P'h 2 . 

If, now, we wish the ratio of the increments, we have 

and, passing to the limiting ratio, 

du 

dx 

and, if we wish the differential of the function u> we have 

du = Pdx, 

du 
or —dx = Pdx. 

ax 

If we represent the increment of the variable by k, and 



dx~~ P; 



22 ELEMENTS OF THE 

the differential coefficient by Q, the difference would be 

represented by 

u' - u = Qk + Q'k 2 

du 
and j^dx = Qdx. 

We may conclude from the above, that if we have the 
difference between two states of a function, as 

u' - u = Ph + P'h*, 
that we can immediately pass to the differential of x\,by 
writing du for x\f— u, substituting dxfor h in the second 
member, and omitting the terms which contain h 2 . 

20. If two functions, u and v } dependent on the same 
variable, are equal to each other, for all possible values 
which may be attributed to that variable, the differentials 
of those functions will also be equal. 

For, suppose x to be the independent variable. We 
shall then have (Art. 15), 

vf - u = Ph + P'h\ 
v'-v = Qh+Q'h 2 , 
in which Q is the differential coefficient of v, regarded as 
a function of a\ 

But, since v! and v' are, by hypothesis, equal to each 
other, as well as u and v, we have 

Ph + P'h* =Qh+ Q'h\ 
or, by dividing by h and passing to the limiting ratio, 

P = Q, 

du dv 

hence > s = s> 



DIFFERENTIAL CALCULUS. 23 

du dv . 

and j- ax = —ax, 

ax dx 

that is, the differential of u is equal to the differential of v. 
21. The reverse of the above proposition is not gene- 
rally true ; that is, if two differentials are equal to each 
other, we are not at liberty to conclude that the functions 
from which they ivere derived are also equal. 

For, let 

u = v± A, 

in which A is a constant, and u and v both functions of 
x. Giving to x an increment A, we shall have 

1/ = v' ± A, 
from which subtract the primitive equation, and we ob- 
tain 

vf — u —v f — v, 

and, by substituting for the difference between the two 

states of the function, we have 

Ph + P'h 2 = Qh+Q'h 2 

Dividing by h, and passing to the limiting ratio, we 

obtain 

_ _ _ du dv 

P = Q: thaus ^ = Tx ; 

. du dv _ 

hence, -7- dx= -^-dx; 

dx ax 

or, what is the same thing, by merely changing the form, 

du = dv. 
Here we see that although v may be greater or less than 
u by the constant quantity A y still its differential will 
always be equal to that of u. 



24 ELEMENTS OF THE 

Hence, also, we conclude that every constant quantity, 
connected with the variable by the sign 'plus or minus, 
will disappear in the differentiation. 

The reason of this is apparent; for, as a constant ad- 
mits of no increase or decrease, there is no ultimate or 
last difference between two of its values ; and this ulti- 
mate or last difference is the differential of a variable 
function. Hence the differential of a constant quantity- 
is equal to 0. 

22. If we have a function of the form 
u = Av, 
in which u and v are both functions of #, and give to x 
an increment A, we shall have 

u' — u — A(v' — v ), 
or Ph + P% = A{Qh+ Qh 2 ) ; 

and, by dividing by A, and passing to the limiting ratio, 

we have 

P = AQ, 

or Pdx = A Qdx. 

But Pdx = du, and Qdx = dv, 

hence, du — Adv ; 

that is, the differential of the product of a variable 

quantity by a constant, is equal to the differential of the 

variable multiplied by the constant. 



DIFFERENTIAL CALCULUS. 25 



CHAPTER II. 

Differentiation of Algebraic Functions — Succes- 
sive Differentials — Taxjlor's and Maclaurinls 
Theorems. 

23. Algebraic functions are those which involve the sum 
or difference, the product or quotient, the roots or powers, 
of the variables They may be divided into two classes, 
real and imaginary. 

24. Let it be required to find the differential of the 
function. 

u = ax. 

If we give to x an increment h, and designate the 
second state of the function by u' ', we shall have 

v! = ax -f- ah =. u + ah, 



= a: 



hence, du = adx, or ~t~^ jX — a ^ x * 

25. As a second example, let us take the function 
u = ax 2 . 



26 ELEMENTS OF THE 

If we give to x an increment h, we have 
v! = ax 2 + 2 tf7*# + ah 2 , 









A 


— igua. —j- U-/& . 


nee 


» 




cZw = 


2aa?c?a?. 


26. 


For 


a third 


example, 
u 


take the function 
= ax 3 : 



giving to x an increment h, we have 



— - — = 3 ax 2 -^ 3 axlx -f ah 2 , 
h 

or passing to the limit 

-j— = 3 ax 2 ; hence, e?w = 3 ax 2 dx. 
ax 

27. Let us now suppose the function u to be composed 
of several variable terms : that is, of the form 

u = y + z — w = f{x\ 

in which y, z, and w, are functions of x. 

If we give to x an increment h, we shall have 

u' —u — {y r — y) 4- (z f — z) — {w r — w) : 
hence, (Art. 19), 

u'-u = (Ph + P r h 2 ) + ( Qh + Qh 2 ) - (Lh 4- I/A 2 ), 
or, v ^ = (P + Fh) + (Q+Q f h)-(L + L'h), 
or by passing to the limit 



DIFFERENTIAL CALCULUS. 27 

and multiplying both members by dx, we have 

—7- dx = Pdx-\- Qdx — L dx. 
ax 

But since P, Q, and L, are the differential coefficients 
of y, z, and iv, regarded as functions of x, it follows (Art. 
18) that, the differential of the sum, or difference of any 
number of functions, dependent on the same variable, is 
equal to the sum or difference of their differentials taken 
separately. 

28. Let us now determine the differential of the product 
of two variable functions. 

If we designate the functions by u and v, and suppose 
them to depend on a variable x, we shall have 

u' = u + Ph + P'h 2 , 
v f = v + Qh + Q'h\ 
and by multiplying 

iiV = (u + Ph+ P'h 2 ) (v + Qh + Qfh*) 

= uv + vPh + uQh + PQh 2 + &c ; 
hence 

u'v' — uv 

= vP+ uQ-{- terms containing h, h 2 , 6c h 3 . 

If now, we pass to the limiting ratio, we have 

d(uv) D , „ 
-\-J- = vP + uQ; 

dx 

therefore, d(uv) = vPdx + uQdx = vdu -f udv. 

Hence, the differential of the product of two functions 
dependent on the same variable, is equal to the sum of the 



28 ELEMENTS OF THE 

products which arise by multiplying each by the differ- 
ential of the other. 

If we divide by uv, we have 

d(uv) du dv 
uv u v ' 
that is, the differential of the product of two functions, di- 
vided by the product, is equal to the su?n of the quotients 
which arise, by dividing each differential by its function. 
29. We can easily determine from the last formula, 
the differential of the product of any number of functions. 
For this purpose, put v = ts, then 

dv _ d(ts) _ dt ds 
v ts t s 

and by substituting for v in the last equation, we have 

d{uts) du dt ds 
uts u t s 

and in a similar manner, we should find 

diutsr . . . . ) du , dt , ds , dr s 

-^ 1 = + — + 4- &c. 

utsr .... u t s r 

If in the equation 

d(uts) _ du dt ds 

r — — I ~ I > 

UtS UtS 

we multiply by the denominator of the first member, we 
shall have 

d(uts) = tsdu + usdt + utds ; 

and hence, the differential of the product of any number 
of functions, is equal to the sum of the products which 



DIFFERENTIAL CALCULUS. 29 

arise by multiplying the differential of each function by 
the product of all the others. 

30. To obtain the differential of any fraction, as 

— we make 
v 

— = t, and hence u — tv. 
v 

Differentiating both members, we find 
du = vdt + tdv ; 
finding the value of dt, and substituting for t its value 

— , we obtain 
v 

, du udv 

dt= r , 

V tr 

or by reducing to a common denominator 

- vdu — udv 

at = g ; 

v 2 

hence, the differential of a fraction is equal to the deno- 
minator into the differential of the numerator, minus the 
numerator into the differential of the denominator, divided 
by the square of the denominator. 

u 

31. If the numerator u is constant in the fraction t = — , 

v 

its differential will be (Art. 21), and we shall have 

, udv dt u 

dt= r , or -=- = =. 

v* dv ir 

When u is constant, t is a decreasing function of v (Art. 
5), and the differential coefficient of t is negative. 

This is only a particular case of a general proposition. 



30 ELEMENTS OF THE 

For, let u be a decreasing function of x. Then, if we 
give to x any increment, as h, we have 

u'=u + Ph+P'h 2 , 

or, u'-u = Ph + P / h 2 . 

But by hypothesis u>u f ; hence, the second member 
is essentially negative for all values of h ; and, passing 
to the" limiting ratio, 

— = - P. 
dx 

hence, the differential coefficient of a decreasing function 

is negative. 

32. To find the differential of any power of a function, 

let us first take che function u n , in which n is a positive 

and whole number. This function may be considered as 

composed of n factors each equal to u. Hence, (Art. 29), 

d(u n ) d{uuuu . . . . ) du . du du du 

= 1 1 1 r 



u n (uuuu . . . . ) u u u u 

But since there are n equal factors in the first member, 
there will be n equal terms in the second ; hence, 

d(u n ) _ ndu 

~1F ~~u~ ; 

therefore, d (u n ) = nu n ~ l du. 

r 
If n is fractional, represent it by — , and make 

r 

v = u % whence, u T = v"; 
and since r and s are supposed to represent entire num- 
bers, we shall have 

ru r ~ l du = sv"~ l dv ; 



DIFFERENTIAL CALCULUS. 31 

from which we find 



, ru r - x , nr 1 , 
dv = — -— r du = du. 



or by reducing 



su 9 



dv — — u s du; 
s 



which is of the same form as the function 
d{u n ) = nu n ~ x du, 

T 

by substituting the exponent — for n. 



Finally, if n is negative, we shall have 

from which we have (Art. 31), 

7/ - n \ j/l\ — d(u n ) —niT~ l du 
d(u ) = *(-) = — ±-> = ^ ; 

hence, by reducing 

d{u~ n ) = — nu~ n ~ 1 du. 

Hence, the differential of any power of a function, is 
equal to the exponent multiplied by the function ivith its 
primitive exponent minus unity, into the differential of the 
function. 

33. Having frequent occasion to differentiate radicals of 
the second degree, we will give a specific rule for this 
class of functions. 



Let v = V^T or 



— ,/2 . 



v — u 



then, dv = — it du = —u du = 



2 2 . 2Vu~' 



32 ELEMENTS OF THE 

that is, the differential of a radical of the second degree, 
is equal to the differential of the quantity under the. sign, 
divided by twice the radical. 

34. It has been remarked (Art. 3), that in an equation 
of the form 

u = f(x), 

we may regard u as the function, and x as the variable, 
or x as the function, and u as the variable. We will 
now show that, the differential coefficient which is obtained 
by regarding u as a function of x, is equal to the recip- 
rocal of that which is obtained by regarding x as a func- 
tion of u. 
If we have 

u =f{x), 
and give to x an increment h, we have (Art. 19), 

u'-u^Ph + P'h*. (1) 
But, if x be expressed in u, and we have 

x =/( w )i 

and then give to u an increment k, we shall nave 

rf-x = h = Qk + Q'k 2 . (2) 

But k = u' — u. Substituting these values for v! — u, 
and h, in equation (l), and we have 

k — PQk + terms containing the higher powers of k. 

Dividing by k, and passing to the limiting ratio, we have 

1 = PQ, or P = ± 



DIFFERENTIAL CALCULUS. 33 

To illustrate this by an example, let 

i_ 
u = or 3 , whence x — y/u — u 3 • 



Now, 




du 
dx 


:3^ = 


2 

3u 3 ; 


bin 


t regarding 


X 


as the function 










dx 
du 


1 -I 

U 3 

3 


1 

— 2 * 

3u T 



35. If we have three variables u, y, and a?, which are 
mutually dependant on each other, the relations between 
them may be expressed by the equations 

u=*f(y), and y=f'(x). 

If now we attribute to x an increment h, and designate 
by k, the corresponding increment of ij, we shall have 
(Art. 19), 

U'= U + Pk + Pk\ i/ = y+Qh+ Q'h\ 

and ^Z± = P-^P% y^=JL=Q+Q?h, 

If we multiply these equations together, member by 
member, we shall have 

u ^ x y^ = (P + Pk)(Q + Qh); 

but k = y r — y ; hence, by dividing and passing to the 
limiting ratio, we have 

du _ du dy 

dx dy dx 

and hence, if three quantities are mutually dependant on 



34 ELEMENTS OF THE 

each other, the differential coefficient of the first regarded 
as a function of the third, will be equal to the differential 
coefficient of the first regarded as a function of the second, 
multiplied by the differential coefficient of the second re- 
garded as a function of the third. 

36. Let us take as an example 



we find 



v = bu 3 , u — ax 2 , 



* 3W, ^ = 2ax. 

du dx 



But, -r— = -=— X -=— = 3 bu 2 x 2ax = 6 abu 2 x ; 

dx du dx 



and by substituting for u 2 , its value a 2 x\ 

dv 

—— = 6a 3 bx 5 , and dv = 6a 3 bx 5 dx. 

dx 



EXAMPLES. 

1. Find the differential of u in the expression 
u= Vd'-x 2 . 

Put a 2 — x 2 = y, then u = y 2 , and the dependence be- 
tween u and x, is expressed by means of y, and u is 
an implicit function of x. Differentiating, we find 



DIFFERENTIAL CALCULUS. 35 

by multiplying the coefficients together we obtain 

du 1 , o *\-4« — x 

= {a 2 — x 2 ) *2x 



dx 2 Va 2 -*?' 

hence, 

xdx 



du = 



Va 2 -a? 

2. Find the differential of the function 

u = (a -f bx n ). 
Place a-\-bx n = y : then u = y m ; and 

dy 



nence, 



-nbx 11 ' 1 ; 
dx 

nil tn— 1 

dx 



du = mnb {a + bx n ) x n l dx. 
3. Find the differential of the function 
u = x(a 2 -f x 2 ) Va 2 — x 2 , 
du=((a 2 + x 2 )V'a T ^^)dx + xVa^^d(a 2 -^x 2 \ 



+ x(a 2 + x 2 )dVa 2 -x 2 , 

in which the operations in the last two terms are only 
indicated. If we perform them, we find 

d(a 2 -f x 2 ) = d(x 2 ) = 2xdx, 

di — x 2 ) —xdx 



d -\/a 2 — x 2 



2^/a 2 -x 2 V^x> 



36 ELEMENTS OF THE 

Substituting; these values, we find 



du = 



(a 2 + x 2 )Va 2 -x 2 + 2x 2 Va 2 -x 2 - 



x 2 (a 2 + x 2 ) 



(Ix 



or, reducing to a common denominator and cancelling the 
like terms, 

_ (a* + a 2 x 2 -4:x*)dx 
Vci 2 — x 2 

4. Find the differential of the function 

tf-x 2 
U ~ tf + aW + x*' 

(a 4 + aV+ x*)d(a 2 - x 2 ) - (a 2 - x 2 )d{a^+ c¥+ a? 4 ) 



du — 



from which we find 



(a 4 +«V+^) 5 



du 



2x{2a i + 2a 2 x 2 -x 4 )dx 



(a 4 + aV + 07 4 ) 2 
5. Find the differential of the function 



M= v / ( a -^ + " J/ ^ E ^) 3 - 



Make 



5 r = V / (c 2 -or J ) 2 , 



then we shall have 



DIFFERENTIAL CALCULUS. 37 

we therefore have (Art. 32), 

3 --i 

du = -—{a — y-\-zY d(a — y-\-z\ 



j{a-y + z) *(-dy + dz), 



_ — 3 dy + Sdz 
4Va — y-\-z 

But from the equations above, we find 

/ b \ d -\/~x~ — bdx 



dz = d(c 2 -a?) z =—{(?- x 2 )* ^(c 2 -* 2 ), 
o 

2 . 9 „.-i ^ _ — 4a?efo? 

= -£■((? — or) 3 x — 2#aa? 



3 V J 3^c 2 -^' 

Substituting these values of dy and cfe, in the ex- 
pression for du, we find 



f 



Sb 4a? 



2a?Va? V^-a? 2 , 

du= <l — ^ y dx. 



1 , — da? 

6. w = — , aw = 5- e 

x x 2 

1 , — rcda? 



38 ELEMENTS OF THE 

„ /- =■ , (a + x)dx 

8. u = V2ax 4- x 2 , du = V ; • - r 

9. M = (a a + 0*) 8 , du = 6(a 2 + a*fxdx. 

10. u = a 6 -f 3aV + 3oV + * 6 , dti = 6(a 2 + arftfcfo. 

11. w = — 7==, dw = 5 . 

Vl-x 2 



12. U 



(1-3*)' 

x , c?a? 



/ _\3 3(g+ V^J ^p 
13. u = [a + V^), ^ = ~ 7=^ • 



14. «=[a + yi- ~J, du — 



g[ a+v /eg]L 



15. w = a? 2 ?/ 2 dw = 2o?ydy + 2y 2 xdx. 

^ Va 2 +^V6 2 +2/ 2 

a? n , nx n ~ 1 dx 

17. W = — — Si C?W 



(1+*)"' "" (l + *) n+1 • 

1 -\-a? , Axdx 

18 . u = T -^ t du r<j=gf 

19 u- ^+y ^ = *(<** +^y) - (g+ y) 3 (fe 

2 s • z* 



DIFFERENTIAL CALCULUS. 39 

21. Find the differential coefficient of 
F(x) = 8x*-3x 3 -5x 





Arcs. 32# 3 -9 < r J — 5. 


22. 


Find the differential coefficient of 




F(x) = (x 3 + a){Sx 2 + b) 




Ans. 1 5 a? 4 + 3 a?b + 6 a#. 


23. 


Find the differential coefficient of 




J F(*) = (a* + * 2 ) 2 , 




Ans. 2 («a? + a? 2 ) (« + 2 #). 


24. 


Find the differential coefficient of 




F(r\ — °° 




a?+ Vl-a? 2 




4 H*. ; ■ . 



Vl-^ 2 (l+2o?Vl-a? 2 )' 

Of Successive Differentials. 

37. It has been remarked (Art. 19), that the differ- 
ential coefficient is generally a function of x. It may 
therefore be differentiated, and x may be regarded as the 
independent variable. A new differential coefficient may 
thus be obtained, which is called the second differential 
coefficient. 



40 ELEMENTS OF THE 

38. In passing from the function u to the first differ- 
ential coefficient, the exponent of x in every term in 
which x enters, will be changed ; and hence, the rela- 
tion which exists between the primitive function u and 
the variable x, is different from that which will exist 
between the first differential coefficient and x. Hence, 
the same change in x will occasion different degrees of 
change in the primitive function and in the first differential 
coefficient. 

The second differential coefficient will, in general, be 
a function of x : hence, a new differential coefficient 
may be formed from it, which will also be a function 
of x ; and so on, for succeeding differential coefficients. 

If we designate the successive differential coefficients 
by 

p, q, r, s, &c, 

we shall have 

du _ dp _ dq _ „ 

dx dx ' dx 

But the differential of p is obtained by differentiating 

du 
its value — , regarding the denominator dx as con- 
ax 

stant : we therefore have 



d 



/du\ , d 2 u , 



and by substituting for dp its value, we have 
d 2 u 



DIFFERENTIAL CALCULUS. 41 

The notation d' 2 u, indicates that the function u has 
been differentiated twice, and is read, second differential 
of u. The denominator doc 2, expresses the square of the 
differential of x, and not the differential of x 2 . It is 
read, differential square of x, or differential of x squared. 

If we differentiate the value of q, we have 



dq; 







«S)- 


-.dq, 


or, 


d 3 u 

dx 2 


hen( 


:e, 




d 3 u 
dx 3 ~ 


= r, 


&c., 


and 


in 


the same manner we 


may 


find 








dx" 


= s, 





d 3 u 
The third differential coefficient -7-^-, is read, third 

dx 6 

differential of u divided by dx cubed; and the differ- 
ential coefficients which succeed it, are read in a similar 
manner. 

Hence, the successive differential coefficients are 

du _ d 2 u _ d 3 u _ d*u _ „ 

Tx~ h dx 1 ' 9, ~d^- r ' l^~ Si &C '' 

from which we see, that each differential coefficient is 
deduced from the one which precedes it, in the same 
way that the first is deduced from the primitive function. 

39. If we take a function of the form 
u = ax n , 



42 ELEMENTS OF THE 

we shall have for the first differential coefficient, 

du „ , 

-7- = nax . 
ax 

If we now consider n, a, and dx } as constant, we 
shall have for the second differential coefficient 



da? 

and for the third, 



z A = n{n-l)ax n 2 , 



d 3 u 

-^ = n(n-l)(n-2)ax n -\ 

and for the fourth, 

^ = n(n - l)(w - 2)(n - 3)aa^- 4 . 

cix 

It is plain, that when n is a positive whole number, the 
function 

u = ax n , 

will have n differential coefficients. For, when n dif- 
ferentiations shall have been made, the exponent of x in 
the second member will be ; hence, the nth differential 
coefficient will be constant, and the succeeding ones will 
be equal to 0. Thus, 

^ = n(n-l)(n-2){n-S) o.l, 

and, j-^0. 



DIFFERENTIAL CALCULUS. 43 

Taylor s Theorem. 

40. Taylor's Theorem explains the method of de- 
veloping into a series any function of the sum or difference 
of two variables that are independent of each other, ac- 
cording to the ascending powers of one of them. 

41. Before giving the demonstration of this theorem, 
it will be necessary to prove a principle on which it de- 
pends, viz : if we have a function of the sum or difference 
of two variables 

u = f(x±y), 

the differential coefficient will be the same if we suppose x 
to vary and y to remain constant, as when we suppose y 
to vary and x to remain constant. 

For, make x±y = z ; 

we shall then have 

u = f{z) 

, du 

and - — = p. 

dz r 

If we suppose y to remain constant and x to vary, 
we have 

dz = dx, 

and if we suppose x to remain constant and y to vary, 
we have 

dz = dy. 

But since the differential coefficient p is independent 
of dz 1 (Art. 15), it will have the same value whether, 

dz = dx, or, dz = dy. 



44 ELEMENTS OF THE 

To illustrate this principle by a particular example, let 
us take 

u = (x + y)\ 

If we suppose x to vary and y to remain constant, 
we find 

and if we suppose y to vary and x to remain constant, 
we find 

Ty = n{cc + y) , 

the same as under the first supposition. 
42. It is evident that the 

f(x+y), 

must be expressed in terms of the two variables x and y, 
and of the constants which enter into the function. 
Let us then assume 

f(x + y) = A + By a + Cy h + Dy c +, &c, 

in which the terms are arranged according to the ascend- 
ing powers of y, and in which A, B, C, D, &c, are inde- 
pendent of y, but functions of x, and dependant on all 
the constants which enter the primitive function. It is 
now required to find such values for the exponents a, b, c, 
&c, and the coefficients A, B, C, D, &c, as shall ren- 
der the development true for all possible values which 
may be attributed to x and y. 



DIFFERENTIAL CALCULUS. 45 

In the first place, there can be no negative exponents. 
For, if any term were of the form 

it may be written 

B 

a ' 

y 

and making y = 0, this term would become infinite, and 
we should have 

f(x) = cc, 

which is absurd, since function of x, which is independent 
of y, does not necessarily become infinite when y = 0. 

The first term A, of the development, is the value 
which the primitive function assumes when we make 
y = 0. If we designate this value by u, we shall have 

f(x)=u. 

If we make 

f(x + y) = u', 

and differentiate, under the supposition that x varies and y 
remains constant, we shall have 

<M_dA dB_ a dC_ b ,dD c . 

dx dx dx dx dx 

and if we differentiate, regarding y as a variable and x 
as constant, we shall find 

^L=aBy a - 1 + bCy h - x + cDy e ~ l +, &c. : 
dy 

But these differential coefficients are equal to each other 
(Art. 41); hence, the second members of the equations 



46 ELEMENTS OF THE 

are equal, and since the coefficients of the series are 
independent of y, and the equality exists whatever be the 
value of y, it follows that the corresponding terms in each 
series will contain like powers of y, and that the coef- 
ficients of y in these terms will be equal (Alg. Art. 244). 
Hence, 

a — 1 = 0, b — l = a, c — 1=6, &c, 
and consequently 

a = l, b = 2, c = 3, &c. ; 

and comparing the coefficients, we find 



B- dA 

dx 


C=- 


1 d f,D= l 

2 dx 3 


dC 
dx 




And since we have made 






/(«)=*= 


u, 


and f(x -f- y) - 


= u', 




we shall have 










du 

A = U, B = — r- 

ax 


, c 




dPu 


1.2. 


Sdx 3 ' 


and consequently, 










. du 
u'=u + -j-y-\- 


d?u 


y 2 { (Pu y 3 


L 


&c. 


dx 2 


1.2 dx 3 1.2.3 T 



43. This theorem gives the following development for 
the function 

u^ix + yf, 

du _ , ePw , ,\ n i o 

a = x , -7— = no?" , -y- 2 - =n(n — 1 )a? , &c. : 
aa? oar 



DIFFERENTIAL CALCULUS. 47 

hence, 

J / \« n 1 V(U— 1) „ 9 o 

w'= (# + 7/) n = a? n 4- nx x y + — -x n ~ 2 y z , 

-D(»-»)^- y+ ; &c . 



1.2.3 

44. The theorem of Taylor may also be applied to the 
development of the second state of any function of the 
form 

u = /(#)> 

when x receives an arbitrary increment h, and becomes 
x -h h. For, if we substitute h for y, we have 

, du, , c? 2 w 7i 2 , d?u Ji 3 , « 

M=M+ ^ + ^r2 + -^rx3 + ' &c - ; 

v! — u du d 2 u h <Pu h 2 

~T~ = ^? dtfUS + ^"17273 + ' &C,J 



/du 2 I cPu h \ 



du 
dx 

Now, it is plain that h may be made so small that the 
, id 2 u 1 d 3 u h , v 

term H^-T¥ + 5?r¥73 + ' &c --) 

shall be less than any assignable quantity, and conse- 

sequently less than — . Then, for any value of h still 
dx 

smaller, we shall also have 

^ > /c^w 1 d 3 u h v 

dx h \dx 2 lT2 + dx^hJ73 +,& ' C ')' i 

or, if we multiply both sides of the inequality by h, we 

shall have 



48 ELEMENTS OF THE 

du, ^ d 2 U Jl 2 d/u Jl 3 

dec "dot* 1.2 ^^1.2.3^' C " 

that is, ivhen a series is expressed in the ascending 
powers of a variable, so small a value may be assigned 
to that variable as shall render the first term of the series 
greater than the sum of all the other terms, and this in- 
equality will increase for all values of the variable which 
are still less. Under such a supposition the sign of the 
series will depend on that of its first term.. 

45. Remark. The theorem of Taylor has been demon- 
strated under the supposition that the for?n of the function 

vf =f(x+ y), 
is independent of the particular values which may be 
attributed to either of the variables x or y. Hence, when 
we make y = 0, and obtain 

this function of x ought to preserve the same form as 
f(x + y) ; else there would be values of x in one of the 
functions, 

u'=f(x+y), u=f(x), 

which would not be found in the other, and consequently 
some of the values of x would be made to disappear 
when a particular value is assigned to y, which is entire- 
ly contrary to the supposition. 
If the function be of the form 

u' — b + Va — x -f y, 
we shall have 

u = b + Va—x. 



DIFFERENTIAL CALCULUS. 49 

If we now make x — «, we shall have 
a 1 —b-\- \Ty^ and u — b, 

in which we see, that u' and u are expressed under dif- 
ferent forms ; and hence, the particular value of y = 
changes the form of the function, which is contrary to the 
hypothesis of Taylor's theorem. When, therefore, the 
function 

v! = J\x + y), 

shall change its form by attributing particular values to 
x or y, the development cannot be made by Taylor's 
theorem, for such particular values. 

46. The particular supposition which changes the form 
of the function will, in general, render the differential 
coefficients in the development equal to infinity. 

If we have 



u' = c + Vf+x — y, 

then, u = c + ^/f-\-x, 

du 1 

d 2 u_ 1 

M 2x2(f+xfi 

d 3 u 1 . 3 

dx " 2x2x2(f+x)T 

&c. &c. 

in which all the coefficients will become equal to infinity 
when we make x = — f. 



50 ELEMENTS OF THE 

47. If we have a function of the form 



v! — b 4- "Vol — x -f y, 

in which n is a whole number, all the differential coeffi- 
cients of u, for x=a will become infinite. For, we have 



hence, 



b+ya 


— X- 


= b + (a- 


-x)\ 


du 


1 


1 




dx 


n , 
(a 


n-l > 

— x) n 




d 2 u 


(1- 


n) 1 




dx 2 ~ 
&c. 


n 2 


(a — x 
&c 


2n-l» 

" 



all of which become infinite when we make x = a. 

Maclauriri's Theorem. 

48. Maclaurin's Theorem explains the method of 
developing into a series any function of a single variable 
Let us suppose the function to be of the form 

u = f{x). 

It is plain that the value of f(x) must be expressed in 
terms of x, and of the constants which enter into f(x). 
Let us therefore assume 

u = A + Bx a -f Cx b + Dx e -f, &c, 

in which the terms are arranged according to the ascend- 
ing powers of x, and in which A, B, C, D } &c., are 



DIFFERENTIAL CALCULUS. 51 

independent of x, and dependent on the constants which 
enter into f(x). 

It is now required to find such values for the exponents 
a, b, c, &c, and the coefficients A, B, C, D, &c, as 
shall render the development true for all possible values 
which may be attributed to x. 

If we make x = 0, u takes that value which the f(x) 
assumes under this supposition, and if we designate that 
value by U we shall have 

U =A. 

The first differential coefficient is 

d -^ = aBx a ~ l + bCx b ~ l + cDx c ~ l + &c, 
ax 

and since this does not necessarily become when we 
make x = 0, it follows that there must be one term in the 
second member of the form x° : hence, 

a— 1=0, or a = 1 ; 

and making x = 0, we have 

4^ = 5= V 

ax 

The second differential coefficient is 

^ = b(b-l)Cx b -* + c(c-l)Dx e - 2 +&c.; 

but since the second differential coefficient does not neces- 
sarily become 0, when x = 0, we have 

6-2 = 0, or 6 = 2: 



52 ELEMENTS OF THE 

hence, by making x — 0, we have 

d 2 u on d 2 u 1 XJ" 

-—— = 2 6, or C = -=-=■ — .= — 

da? dx 2 2 1.2 

We may prove in a similar manner that 

3 u 1 V" 



c-3 and D 



dx? 1.2.3' 1.2.3 



Having designated by U what the function becomes 
when we make x = 0, and by t/ 7 , Z7", U ,r , &c, what 
the successive differential coefficients become under the 
same supposition, we shall have 

/(*) = U+ Ux + 0" — + IP" j£- + &c. 
^ w 1.2 1.2.3 

49. The theorem of Maclaurin may be deduced imme- 
diately from that of Taylor. 
In the development 

, , du d 2 u y 2 , dhi if c 

dx dx 2 1 .2 dx 3 1 .2.3 

the coefficients a, — - — , -^-5, &c, 

dx dx z 

are functions of a?, and also dependent on the constants 
which enter into f(x + y). 

If we make x = 0, the /(a? + y) becomes /(y), and 
each of the differential coefficients being thus made inde- 
pendent of x } will depend only on the constants which 
enter into f(x + y), and which also enter into f(y). 
Hence, if we designate by 

U, U ; , V", U'", U"", &c, 



DIFFERENTIAL CALCULUS. 53 

the values which the coefficients assume under this 
hypothesis, we shall have 

/(y) = U+ U'y+ U"fL + U'" Y g 1 + W'^jL-^+Scc. 

50. If we take a function of the form 
« = (a + x) n , 



we shall have 



£ = »(« + *) , 



g = n(n-l)(a + *)-«, 

&c. = &c. 

which become, when we make x = 0, 

U=a n , V' = na n ~\ V" = n(n-\)a n -\ &c; 
hence, 

{a + x) n = a n + nd— 1 * + "fo"" 1 ^ — V + &c. 

l . <4 

51. Remark 1. The theorem of Maclaurin has been 
demonstrated under the supposition that the f(x) reduces 
to a finite quantity when we make x — 0. The case, 
therefore, is excluded in which x — renders the function 
infinite. Thus, if we have 

u = cot x, u — cosec x, or u = log x, 

and make x = 0, we find w = ao ; hence, neither of these 
functions can be developed by the theorem of Maclaurin. 



54 ELEMENTS OF THE 

Remark 2. We have already seen (Art. 45.), that the 
theorem of Taylor does not apply to those cases in 
which the form of the function is changed by attributing 
a particular value to one of the variables : the theorem 
therefore fails for particular values, but is true for all 
others, and hence, the general development never fails. 

In the theorem of Maclaurin the failure arises from the 
form of the function : hence, it is the general development 
which fails, and with it, all the particular cases. 

EXAMPLES. 

1. Develop into a series the function 

i 

2. Develop into a series the function 

u= ^/(a 2 -x 2 f=a^(l-^j\ 

3. Develop into a series the function 

a + x \ a J 

4. Develop into a series the function 



DIFFERENTIAL CALCULUS. 55 



CHAPTER III. 

Of Transcendental Functions. 

52. If we have an equation of the form 

u = a x , 

in which a is constant, it is plain that u will be a function 
of x ; and if a be made the base of a system of logarithms, 
x will be the logarithm of the number u (Alg. Art, 257). 
When the variable and function are thus related to each 
other, u is said to be an exponential or logarithmic func- 
tion of x. (Art. 9). 

53. The functions expressed by the equations 

u = sin x, u = cos x, u = tang x, u= cot x, &c, 

are called circular functions. 

The logarithmic and circular functions are generally 
called transcendental functions, because the relation be- 
tween the function and variable is not determined by the 
ordinary operations of Algebra. 

Differentiation of Logarithmic Functions. 

54. Let us resume the function 

u = a x . 



56 ELEMENTS OF THE 

If we give to x an increment A, we have 

and u'— u = a** h -a x = a x (a h -l). 

In order to develop a h , let us make a = l+b, we shall 
then have 

a ^ (1+6) ^ 1+ A 6+ ^z±) 62+ ^- 1 )(^- 2 V +&c; 

K J 1 1.2 T 1.2.3 ' 

hence, 

^ 1= Aa+ £a=D.y+ »<»-D(»-»y + &c „ 

1 1.2 1.2.3 

_ , fc {h-l)b 2 (h-l)(h-2)b 3 



h 



Vl + 1 2 + L2 T + C 'J> 



from which we see, that the coefficients of the first power 
of h will be 

Vl 2 3 /' 

replacing b by its value a — 1, and passing to the limit, 
we obtain 

cfo dx \ 1 2 3 y 

or if we make 

ft = £ - i _(«-l£ + (a-l£_ &C) 

1 2 o 

— — = /ca*, or da x = fta'cfo ; 
dx 

in which /c is dependent on a. 



DIFFERENTIAL CALCULUS. 57 

The successive differential coefficients are readily found. 

For we have 

da x 



d 



dx 
da 



a x k, 



(-T-) — da x k = a x k 2 dx ; 

\ (XX ' 



hence, -7-^- = <fk 2 , 

— -a x k 3 
dx> ~ ak > 

&c. &c 

d n a x 7 _ 

= a x k n . 



dx n 



55. It is now proposed to find the relation which exists 
between a and k. For this purpose, let us employ the 
formula of Maclaurin, 

u = /(*) = U+ V± + U»JL + U'"^-^ + &c. 

If in the function 

u = a x , 

and the successive differential coefficients before found, 
we make x = 0, we have 

17=1, U'=k, U'=k\ V"'=k\ &c; 
hence, 

_ H KX K XT K X: 

a=1 + T + T2 + TsT3 + &c - 



If we now make x = -=-, we shall have 



a " = 1 + T + T2 + Th + &c - : 



58 ELEMENTS OF THE 

designating by e the second member of the equation, and 
employing twelve terms of the series, we shall find 

e = 2.7182818; 

hence, a k —e, therefore a = e k . 

But, 2.7182818 is the base of the Naperian system of 
logarithms (Alg. Art. 272) ; hence, the constant quantity 
k is the Naperian logarithm of a. 

By resuming the result obtained in Art. 54, 

da x = a x k dx, 

we see that the differential of a quantity obtained by 
raising a constant to a power denoted by a variable ex- 
portent, is equal to the quantity itself into the Naperian 
logarithm of the constant, into the differential of the 
exponent. 

56. If now we take the logarithms, in any system, of 
both members of the equation 





e k = 


a, 






we shall have 












kle- 


: la, or 


k-- 


la 
~7? 


whence, 












da x = 


= ka x dx = 


la 


z x djc ; 


or by recollecting 


that 








we have 


u- 

du 
dx 


= a x , 

la _ 
-Te*' 







DIFFERENTIAL CALCULUS. 59 

or, if we regard x as the function, and u as the variable, 
we have (Art. 34), 

dx __ le 1 
du la a x ' 

Let us now suppose a to be the base of a system of 
logarithms. We shall then have a?= the logarithm of 
u, la = l, and le — the modulus of the system (Alg. 
Art. 272); and the equation will become 

d(lu) = le — , 
u 

that is, the differential of the logarithm of a quantity is 
equal to the modulus of the system into the differential of 
the quantity divided by the quantity itself 

57. If we suppose a = e the base of the Naperian 
system, and employ the usual characteristic V to desig- 
nate the Naperian logarithm, we shall have 

d(l'u)=z — ; 

u 

that is, the differential oj the Naperian logarithm of a 
quantity is equal to the differential of the quantity divided 
by the quantity itself. 

The last property might have been deduced from the 
preceding article by observing that the modulus of the 
Naperian system is equal to unity. 

58. The theorem of Maclaurin affords an easy method 
of finding a logarithmic series from which a table of 
logarithms may be computed. If we have a function of 
the form, 

u = f{x) =z /a\ 



60 ELEMENTS OF THE 

we have already seen that the development cannot be 
made, since fix) becomes infinite when x = (Art. 51.) 
But if we make 

w=/(a?) = Z(l+aO, 

the function will not become infinite when x = ; and 
hence the development may be made. 
The theorem of Maclaurin gives 

u = f{x) =U+U — +U"—+ U" f -^— + &c. 
J v } 1 1.2 1.2.3 

If we designate the modulus of the system of the loga- 
rithms by A, we shall have 



d i=- A (rh?=- M1+ * r2 



dx 3 (1 + ^) 3 



3 = 2A 77—^3 = 2.4(1 + x)~\ 



If we now make x = 0, we have 



U=0, U = A, U f '=-A, V" = 2A, &c. ; 
hence, 

This series is not sufficiently converging, except in 
the case when x is a very small fraction. To render the 
series more converging, substitute — x for x : we then have 



DIFFERENTIAL CALCULUS. 61 

and by subtracting the last series from the first, we obtain 

i( 1+I )- ;( ,-,) = !£) = ^44 + & c.) 

If we make 

1 -\-oc z . z 

= 1 H , we have x = 



1 — x n ' 2n-\- z 

and by observing that 

'(i+3*'(*± £ H<«+«>-* 

we have 
l(n+z)—ln=2A 



2n + z 






from which we can find the logarithm of n -f- z when the 
logarithm of n is known. This series is similar to that 
found in Algebra, Art. 270. 

If we make n = i, and z = 1, we have Zl = 0, and 

'^(l + ^+sV &c -) 

If we make the modulus A = 1 , the logarithm will be 
taken in the Naperian system, and we shall have 

Z'2 = 0.693147180, 
2/' 2 = Z' 4=1. 3862943601 ; 

and by making 2 = 4, and n = 1 , we have 
Z'5 = 1.609437913, 

and /' 2 + Z'5 = Z'10 = 2.302585093. 



62 ELEMENTS OF THE 

If we now suppose the first logarithms to have been 
taken in the common system, of which the base is 10, we 
shall have, by recollecting, that the logarithms of the same 
number taken in two different systems are to each other 
as their moduli (Alg. Art. 267), 

Z10 : no :: A : 1, 

or, 1 : 2.302585093 : : A : 1 ; 

whence, A = = 0.434284482. 

2.3025b509 

Remark. To avoid the inconvenience of writing the 
modulus at each differentiation (Art. 56), the Naperian 
logarithms are generally used in the calculus, and when 
we wish to pass to the common system, we have merely 
to multiply by the modulus of the common system. We 
may then omit the accent, and designate the Naperian 
logarithm by I. 

59. Let us now apply these principles in differentiating 
logarithmic functions. 

1. Let us take the function u = l( . -= ]. 
Make 



Va 2 +a?' 

dz 
and we shall have du = — , 

z 

x 2 dx 



dx \/a 2 + x 2 



VcP+x 2 a 2 dx 

but dz=— — -- — 3 =7 rr! 



*+* (a>+^ 



whence, du = 



DIFFERENTIAL CALCULUS. 63 

d'dx 



x{a 2 -\-x 2 ) 



2. Take the function 



LVT+x— Vi—x _!' 



and make yT+#+ ^/l—x = y, -y/l -\-x— ^/l—x = z, 
which gives 

u = I (—) ~ly — Iz, and du = -^ . 

\ z / y z 

But we have 



, _ dx dx — dx I r—r — ri \ 

y ~2r7i^~2^7\^~2V\^x f ^ * '' 



Vi+x 2v\-x 2Vi-^ /■' 



c&c . dx dx 

az 



ydx 

Whence, 

dy Jz _ zdx ydx 

V z ~ 2yVl^x r ~2zVu^x*' 

-{y 2 +z 2 )dx m 
2yzVl — x 2 

and observing that y 2 -\- z 2 — 4 and yz = 2x, 

we have cfo = , 

a?Vl — a 2 



64 



ELEMENTS OF THE 



3. u = l(x^ Vl + x 2 ), du = 

4 



dx 



Vl + x 2 

- u = -7=- l (x v^r+ Vi-ic 2 )* ^ = 



5. u = l 



6. w = Z 



Vl + a? + x 


i 

~2 


Vl + a?-x-j 




<\Ja + x-\- ya — x 



du 



Vl-x 2 ' 
dx 



du = 



Vl + x 2 
adx 



Va + x— Va — xJ x Vat—x 2 

60. Let us suppose that we have a function of the form 

u = (lx)\ 
Make Ix = z, and we have 

u = z n i du = nz"~ l dz, 

and substituting for z and dz their values, 

d(lx) n = n{lx)n ~ l dx. 
x 

61. Let us suppose that we have 

u = l(lx). 
Make Ix = z, and we shall have, 



, , dz , dx 

u = Iz, du = — , dz = — ; 

Z X 



hence, 



du = 



dx 
xlx 



DIFFERENTIAL CALCULUS. 65 

62. The rules for the differentiation of logarithmic func- 
tions are advantageously applied in the differentiation of 
complicated exponential functions. 

1 . Let us suppose that we have a function of the form 

u = z y , 

in which z and y are both variables. 

If we take the logarithms of both members, we have 

lu = ylz ; 

, du 7 7 dz 

hence, — = dy iz + y — ; 

u z 

or, du — ulzdy -f uy — , 

or by substituting for u its value 

du = dz v = z y lzdy + yz y ~ l dz. 

Hence, the differential of a function which is equal to 
a variable root raised to a power denoted by a variable 
exponent, is equal to the sum of the differentials which 
arise, by differentiating, first under the supposition 
that the root remains constant, and then under the sup- 
position that the exponent remains constant (Arts. 55, 
and 32). 

2. Let the function be of the form 



Make, b z —y, and we shall then have (Art. 55), 

u = a y , du = a y lady ; but dy = b*lbdx, 

X 

hence, du = a b b'lalbdx. 

5 



66 ELEMENTS OF THE 

3. Let us take as a last example 

u ="ar, 

in which z, t, and s, are variables. 
Make, t* = y, we shall then have 

u — z v , du = z y Izdy + yz y ~ l dz. 

But dy = fltds + s f~ l dt ; 

hence, du = z f lz(fltds + sf~ l dt) + f/- l dz 

du = z*f(ltlzds + s M+^. 



Differentiation of Circular Functions. 

63. Let us first find the differential of the sine of an 
arc. For this purpose we will assume the formulas (Trig. 
Art. XIX), 

sin a cos b + sin b cos a 



sin (a + b) = 
sin (a — 6) = 



R 

sin a cos b — sin b cos a 



R 

If we subtract the second equation from the first, 

2 sjn b cos a 



sin (a + 6) — sin (a — 6) = 



R 



and if we make a-\-b = x + h, and a — b = x, we shall 
have 

2 sin — ^cos (a?H h) 

sm (a? + ") — sino: = — , 



DIFFERENTIAL CALCULUS. 

and dividing both members by //, 

, _. . 2 sin — hcos(x-\ h) 

x + h) — smx 2 \ 2 / 



sin (x + h) 



hR 



sin— A cos (aH h) 



R 



67 



If we now pass to the limit, the second factor of the 

cos X 
second member of the equation will become 



In relation to the first factor 



sin — h 
2 



R 



its limit will be unity. 



r, i?sina , sin a cos a 

ror, tang a = , whence = • 

cos a tang a R 

Now, since an arc is greater than its sine and less than 
its tangent* 

sina . , j sina . sina 
< 1, and > 



a tang a 



* The arc DB is greater than a straight line 
drawn from D to B, and consequently greater 
than the sine DE drawn perpendicular to JIB. 

The area of the sector ABD is equal to 

-^BX BD, and the area of the triangle ABC 
2 

is equal to — AB X BC. But the sector is less 

than the triangle being contained within it : hence. 




consequently, 



ABXBD<-ABXBC, 
BD < BC. 



E B 



68 ELEMENTS OF THE 

hence, the ratio of the sine divided by the arc is nearer 
unity than that of the sine divided by the tangent. But 
when we pass to the limit, by making the arc equal to 0, 
the sine divided by the tangent being equal to the cosine 
divided by the radius, is equal to unity : hence the limit 
of the ratio of the sine and arc, is unity. 

When therefore we pass to the limit by making h = 0, 
we find 

dsinx _ cosa? 

~chr~~R~ : 

cosxdx 



hence. dsinx = 



R 



64. Having found the differential of the sine, the diffe- 
rentials of the other functions of the arc are readily de- 
duced from it. 

cos# = sin (90° — x\ dcosoc = <isin(90° — a?), 

and by the last article, 

<2sm(90° - x) = — cos(90° - x) d{90° - x), 
It 



= -—cos(90°—x)dx: 
K 



, , sin a? (fa: 

hence, a cos x = — — ; 

it 



the differential of the cosine in terms of the arc being 
negative, as it should be, since the cosine and arc are 
decreasing functions of each other (Art. 31.) 



DIFFERENTIAL CALCULUS. 69 

65. Since the versed sine of an arc is equal to radius 
minus the cosine, we have 



a ver-sm x = d(R — cos x) = — . 

R 

R, sin x 
66. Since tang x = , we have (Art. 30), 

COS X 



j R cosxd sina: — R s'mxd cosa? 
a tang x = - 



(cos 2 j?+ sin 2 x)dx 

— 2 J 

cos^a? 
but cos 2 a? + sin 2 a? = R 2 : 

R 2 dx 



hence, d tang a? 



R 2 

67. Since cot# = , we have 

tang x 

, M R 2 d tang x R*dx 

d cota? = ^— = = 5— ; 

tang^a? tangpr cos^a? 



9 R 2 sin 2 a? 
but, tang^a? = — 



hence, d cot a? = — 



• 2 > 

sura? 



which is negative, as it should be, since the cotangent is a 
decreasing function of the arc. 



70 


ELEMENTS OF 


THE 


68. 


R 2 

Since sec a? = , 

cos a? 


we have 




, R 2 d cosa? 


R sinxdx 




cos^a? 


cos 2 a? 


but, 


R sin x , 

= tang x, and 

cos X 


R 2 

= s 

cosa? 


hence, 


7 seca? 
a secx = 


tanga?c?a? 
R 2 


69. 


jR 2 

Since cosec x — — : , 


we have 



sec x ; 



d cosec x 



sin a? 
R 2 d. sin a? R cos x dx 



, , cosec a? cot x dx 
hence, a cosec a? = 



R 2 

70. If we make R=l, Arts. 63, 64, 65, 66, 67 r 
will give, 

d sina? = cosa?c?a? (1), 

dcosx= — sinxdx (2), 

d ver sin a? = sin a? da? (3), 

dtanga? = — 4- (4), 

cosa? 

t dx , m . 

a cot a? = nr (5). 

snfa? 

The differential values of the secant and cosecant are 
omitted, being of little practical use. 

71. In treating the circular functions, it is found to be 
most convenient to regard the arc as the function, and the 



DIFFERENTIAL CALCULUS. 71 

sine, cosine, versed-sine, tangent, or cotangent, as the 
variable. If we designate the variable by w, we shall 
have in (Art. 63) sin x = u, and 

, Rdu Rdu 

ax = 



cos a? VR 2 -u 2 ' 
If we make cosx = u, we have (Art. 64), 
Rdu Rdu 



dx 



sin a? VR 2 —u* 

If we make ver-sina? = u, we have (Art. 65), 
Rdu 



dx 



sma? 



But, sina? = ^R 2 — cos 2 a?, and cosx=R — u, 

therefore, cos 2 a? = R 2 — 2 Ru -f u 2 , 



hence, sin x = V2 Ru — w 2 , 

Rdu 



and consequently, c?a? = 



V2#w-w 2; 
If we make tang x = u, we have (Art. 66) 
cos 2 x du 



dx= 



R 



2 » 



, cos a? R , cos 2 a? R 2 R 

but — =r— = , hence 



R sec a? ' i? 2 sec 2 a? # 2 + tangV 

, '.. R 2 du 

hence, - **=-# + *• 



1% ELEMENTS OF THE 

Now, if we make R = l, the four last formulas 
become 

, du , du 

ax = — . , ax = 



Vl - u 2 ' Vl-U 2 ' 



j du , du 

dx = — . dx 



^2u-u 2 1+u 2 ' 

and these formulas being of frequent use, should be care- 
fully committed to memory. 

72. The following notation has recently been introduced 
into the differential calculus, and it enables us to designate 
an arc by means of its functions. 

sin _1 w= the arc of which u is the sine, 

cos~ 1 w= the arc of which u is the cosine, 

tang _1 w= the arc of which u is the tangent, 
&c. &c. &c. 

If, for example, we have 

du 



x =. sin *«, then, dx = 



Vl-u 2 



73. We shall now add a few examples. 
1 . Let us take a function of the form 





a? = cos# ,in *. 


Make 


cos x = z y and sina? = y ; 


then, 


u = z\ and (Art. 62); 




du = z y Izdy + yz v ~ x dz: 



DIFFERENTIAL CALCULUS. 73 

also, dz = — smxdx f and dy = cosxdx * 

hence, du = z y flz dy + — dz \ 

/ sin x \ 

= cosaf na: ( 7cosa?cosa? \dx. 

\ cosa? / 

2. Differentiate the function 

mdu 



Vl-mV 
3. Differentiate the function 



x = cos 1 lu VI — w 2 ) 



da? 



(-1 + 2u 2 )du 



V(l~w 2 4-w 4 )(l— w 2 ) 

4. Differentiate the function 

2c?w 



a? = tang l — , dx — 



2' 4 + w 2 ' 

5. Differentiate the function 

x = sin" 1 (2w Vl — u 2 ), dx = — 

\ / Vl-w 5 

6. Differentiate the function 



, a? 7 ydx — xdv 
u = tang" x — , du — + — = ^-. 

74. We are enabled by means of Maclaurin's theorem 
and the differentials of the circular functions, to find the 



74 ELEMENTS OF THE 

value of the principal functions of an arc in terms of the 
arc itself. 

Let u — f(x) = sin.T : then, 

du d 2 u <Pu 

-r- = cosx, ^-^——smx, — —=— cosa:, 

dx dxr dx* 

d/u . d 5 u 

— — - = smx, -r-r- = 4- cos x. 

dx* dxr 

If we now render the differential coefficients independent 
of x, by making x = 0, we have (Art. 49), 

U=0, 17 = 1, U" = 0, U ! "=z-\, 

17""= Q, U""'-h= + l: 

X X X 

hence, sin x = 1 &c. 

1 1.2.3 1.2.3.4.5 

75. To develop the cosine in terms of the arc, make 

u = f{x) = cos x ; then, 

du . dru dru 

— =— smx, -— = — cosa?, —r^ = sin a?, 

dx dxr dx s 

dru d 5 u 

— = cos*, ^=— m* 

and rendering the coefficients independent of x, we have 
tf=l, tf'=0, U"=-\, c/ w =o, 
CT W =1, ?F'"=0: 

hence, cosa: = 1 - — H ^ - &c. 



DIFFERENTIAL CALCULUS. 75 

The last two formulas are very convenient in calculating 
the trigonometrical tables, and when the arc is small the 
series will converge rapidly. Having found the sine and 
cosine, the other functions of the arc may readily be 
calculated from them. 

76. In the two last series we have found the values of 
the functions, sine and cosine, in terms of the arc. We 
may, if we please, find the value of the arc in terms of 
any of its functions. 

77. The differential coefficient of the arc in terms of 
its sine, is (Art. 71), 

du Vl-u 2 V * 

developing by the binomial formula, we find 

dx , , 1 2 , 1.3, /4 , 1.3.5 , , , 
-t- = 1 +— u 2 + 7r-: u + n A u 6 -f- &c. 
du 2 2.4 2.4.6 

In passing from the function to the differential coeffi- 
cient, the exponent of the variable in each term which 
contains it, is diminished by unity ; and hence, the series 
which expresses the value of x in terms of u, will contain 
the uneven powers of w, or will be of the form 

x=:Au + Bu 3 + Cu 5 + Du 7 + &c; 

and the differential coefficient is 

d 4- = A + SBu 2 + 5 Cu* + 7Du 6 + &c. 
du 



76 ELEMENTS OF THE 

But since the differential coefficients are equal to each 
other, we find, by comparing the series, 

Amh B= i_ t C= _L^, D= 1-3-5 



2.3' 2.4.5' 2.4.6.7' 

hence, 

u , 1 u 3 l.Su 5 , 1.3.5 7 , - 

1 2 3 2.4.5 2.4. G. 7 T 

If we take the arc of 30°, of which the sine is — 

2 
(Trig. Art. XV), we shall have 

ono 1 . 1 1.3 1.3.5 , - 

arc 30° = H ~\ — -f &c; 

2 2.3. 2 3 2.4.5. 2 5 2.4.6.7.2 7 

and by multiplying both members of the equation by 6, 
we obtain the length of the semi-circumference to the 
radius unity. 

78. To express the arc in terms of its tangent, we have 
(Art. 71), 

dx 1 , 2 -i 

Tu = -TT^ = {1 + U) ' 

which gives 

du 
hence the function x must be of the form 

a? = Au + Bu 3 + Cu 5 + Dw 7 , 

and consequently 

^ = A + 3£w 2 + 5 Cu 4 + 7Dw 6 ; 
du 



DIFFERENTIAL CALCULUS. 77 

and by comparing the series, and substituting for A, B, C, 
&c, their values, we find 

_, u U 3 ll 5 u 7 . 

x = taiifif u = h &c. 

& 1 3 5 7 

If we make x — 45°, u will be equal to 1 ; hence, 

arc 45° = 1 - — + L+ &c. 

3 5 7 

But this series is not sufficiently convergent to be used 
for computing the value of the arc. To find the value 
of the arc in a more converging series, we employ the 
following property of two arcs, viz. : 

Four times the arc whose tangent is — , exceeds the 

5 

arc of 45° by the arc whose tangent is — — *. 



* Let a represent the arc whose tangent is Then (Trig. Art, 

XXVI), 5 

2 tan£ a 5 

tang 2 a = - £—-=-—., 

s 1 — tang 2 ** 12 

2 tang 2 a 120 

tang 4a== 2_ — — — _. 

1— tang 2 2a 119 

The last number being greater than unity, shows that the arc 4 a ex- 
ceeds 45°. Making 

4a = A, 45° = ^, 

the difference, 4 a — 45° = A — B = b, will have for its tangent 

tang6 = tangM-l?) = tang^-tang* * . 

8 sv ' 1 -j- tang A tang B 239' 

hence, four times the arc whose tangent is « — , exceeds the arc of 45° by an 

5 
arc whose tangent is — — , 
5 239 



78 
But 



ELEMENTS OF THE 



tang 



x 1 _ 1 



5 3.5 3 ^5.5 5 7.5 7+ ' 



tang 
hence, 



, 1 



1 + ' 



1 



239 239 3(239) 3 ' 5(239) 5 7(239) 7 *' 



arc 45° = < 



a(±- — JL ] \ 
U 3.5 3 + 5.5 3 7.7 7 V 



(. 



1 



+ 



J39) 5 7(239) 7 + J 



\239 3(239) 3 ' 5(239) 5 7(239) 7 
Multiplying by 4, we find the semi-circumference 



= 3.141592653. 



DIFFERENTIAL CALCULUS. 79 



CHAPTER IV. 



Development of any Function of two Variables 
— Differential of a Function of any number 
of Variables — Implicit Functions — Differential 
Equations of Curves — Of Vanishing Fractions. 

79. We have explained in Taylor's theorem the method 
of developing into a series any function of the sum or dif- 
ference of two variables. 

We now propose to give a general theorem of which 
that is a particular case, viz : 

To devehp into a series any function of two or more 
variables, when each shall have received an increment, 
and to find the dijferential of the function. 

80. Before making the development it will be necessary 
to explain a notation which has not yet been used. 

If we have a function of two variables, as 

u =/(*> y)> 

we may suppose one to remain constant, and differentiate 
the function with respect to the other. 

Thus, if we suppose y to remain constant, and x to 
vary, the differential coefficient will be 



80 ELEMENTS OF THE 

and if we suppose x to remain constant and y to vary, 
the differential coefficient will be 

g = /"(*,*). (2). 

The differential coefficients which are obtained under 
these suppositions, are called partial differential coef- 
ficients. The first is the partial differential coefficient 
with respect to x, and the second with respect to y. 

81. If we multiply both members of equation (1) by 
dx, and both members of equation (2) by dy, we obtain 

— dx=z f (x, y) dx, and —dy — f" [x, y) dy. 
ax cky 

The expressions, 

du , du , 

E**' E rfy ' 

are called partial differentials; the first a partial diffe- 
rential with respect to x, and the second a partial diffe- 
rential with respect to y : hence, 

A partial differential coefficient is the differential co- 
efficient of a function of two or more variables, under 
the supposition that only one of them has changed its 
value : and, 

A partial differential is the differential of a function 
of two or more variables, under the supposition that only 
one of them has changed its value. 

82. If we differentiate equation (1) under the suppo- 
sition that x remains constant and y varies, we shall have 



< 



dy 



DIFFERENTIAL CALCULUS. 81 

and since x and dx are constant 



j(du\ _ d(du) 
\dxJ dx 



which we designate by 

d 2 u m 
dx ' 

hence, ^L =/'"(*, y). 

The first member of this equation expresses that the 
function u has been differentiated twice, once with respect 
to x, and once with respect to ?/• 

If we differentiate again, regarding x as the variable, 
we obtain 

which expresses that the function has been differentiated 
twice with respect to x and once with respect to y. And 
generally 

d n+m u 

dx n dy m ' 

indicates that the function u has been differentiated n + m 
times, n times with respect to x, and m times with respect 
to y. 

83. Resuming the function 

if we suppose y to remain constant, and give to x an arbi- 
trary increment h, we shall have from the theorem of Taylor, 

/.,,,* , du h , d 2 u h 2 , d 3 u h 3 B 

j\ -r ,y) ^ dx I da? 1.2 da? 1.2.3 

6 



82 ELEMENTS OF THE 

, . , du d 2 u d 3 u 

mwhlch ' "' £• d?> 1?' 

are functions of x and y, and dependent on the constants 
which enter the f(x,y). 

If we now attribute to y an increment k, the function 
u, which depends on y, will become 

, du 1 , <i 2 w /c 2 , cPw A; 3 , c 

and the function -7- will become 
dx 

du d?u k , d 3 u k 2 dhi k 3 c 

dx dxdy 1 dxdy 2 1.2 dxdy 3 1.2.3 ' 

d?u 
and the function -7-^-, will become 
dxr 

cPu , d 3 u k d*u k 2 d 5 u k 3 , e 

I L (Vc 

dx 2 ^ dx 2 dy 1 T dtfdy 2 1 .2 ^ <fo 2 cfy 3 1.2.3 '' 

d 3 u 
and the function —tt> w ^ Decome 

(Pn d*M k dhi k 2 d*u A 8 & 

da? + dx 3 dy 1 + d^dy 2 1.2 + c^rfy 3 1.2.3 + '' 

&c. &c. &c. &c. 

Substituting these values in the development of 
f(x + ft, y), 



DIFFERENTIAL CALCULUS. 83 

and arranging the terms, we have 

J dy 1 dy 2 1.2 di/l.2.3 

dull (Pu hk d?u hk 2 . 

. <Pu h 2 , <Pk A 2 ft 



eta 2 1.2 da?dy 1.2 

+ da* 1.2.3 +&C - ; 

which is the general development of a function of two 
variables, when each has received an increment, in terms 
of the increments and differential coefficients. 

84. If we transpose u =f(x, y) into the first member, 
and apply the result of Art. 19 to a function of two vari- 
ables, we find 

7r n \n dll dli 

d[f{x>y)} = du - —dx + Wy. 

The differential of f{x, y) — du, which is obtained under 
the supposition that both the variables have changed their 
values, is called the total differential of the function. 

85. If we have a function of three variables, as 

u = f(x, y, z\ 

and suppose one of them, as z> to remain constant, and 
increments h and k to be attributed to the other two, the 
development of f (x+h,y + k, z) will be of the same 
form as the development of / (x -\- h, y + k) ; but u and 
all the differential coefficients will be functions of z. 



84 ELEMENTS OF THE 

If then an increment / be attributed to z, there will be 
four terms of the development of the form 

du , du 7 du , 

w » T~ A » T"^' ^" / * 
cte ay dz 

If w were a function of four variables, as 

u = f(x, y, z, s), 

there would be five terms of the form 

du 1 du 7 du 1 du 

u, — h, —k, — /, — m; 
ax ay dz ds 

and a new variable introduced into the function, would 

introduce a term containing the first power of its increment 

into the development. 

If we transpose u into the first member, and make the 

same supposition as in the last article, we shall have 

du du du 

d[f(x,y,z)-} = Tx dx + ^d y + Tz dz, 

and, for like reasons, 

_ . _ du du _ du du 

d\J(*, V, z, «)] = jjx + Ty dy + Tz dz + T$ ds, 

from which we may conclude that, the total differential 
of a function of any number of variables is equal to the 
sum of the partial differentials. 

86. The rule demonstrated in the last article is alone 
sufficient for the differentiation of every algebraic function. 
1 . Let u = x 2 + y 3 — z ; then 

du 

— dx — 2xdx, 1st partial differential ; 

dx 



DIFFERENTIAL CALCULUS. 85 

du 

-—dy = 3y 2 dy, 2d partial differential ; 

^dz=-dz, 3d 
dz 

hence, du = 2xdx + 3y 2 dy — dz. 

2. Let u = xy ; then, 

du 7 , 

—~dx = y ax, 
ax 

du 7 , 

— ay — xay: 

hence, du = ydx + x dy. 

3. Let u = x m y n ; then, 

— dx = mx m ~ l y n dx, 
dx 



— dy = ny n ~ l x m dy : hence, 

du = mx m - l y n dx + ny n ~ l x m dy = x m ~ l y n ~ l {mydx + nxdy). 

x 
4. Let u = — ; then, 

y 

du 7 dx 
— dx = —, 
ax y 

du j xdy 

dy y y 2 

. 7 ydx — xdy 
hence, du = z 5 — -. 



96 ELEMENTS OF THE 

5. Let u = . y = ay (x 2 -f y 2 )" T ; then, 

Va?+y 2 

du , ayxdx 
-r-dx= 2 1, 

du, ady a y 2 dy 

-rdy= - j; 

a y (o^ + y 2 ) 2 {x 2 + y 2 Y 

, , ayxdx — aa^dy 
hence, du = - — -. 

(3? + y 2 y 

6. Let u — xyzt ; then, 

du = yztdx -\-xztdy + xytdz -+- xyzdt. 

7. Let u — zv; then, 

~dy=.z y lzdy (Art. 55), 
dy * 

~dx = yz^ x dz (Art. 32). 
ax 

hence, du = z^lzdy + yz y ~ l dz. 

Remark. In chapter II, the functions were supposed 
to depend on a common variable, and the differentials were 
obtained under this supposition. We now see that the dif- 
ferentials are obtained in the same manner, when the func- 
tions are independent of each other, and unconnected with 
a common variable. 

87. We have seen (Art. 39), that a function of a single 
variable has but one differential coefficient of the first 
order, one of the second, one of the third, &c. ; while a 



DIFFERENTIAL CALCULUS- 87 

function of two variables has two differential coefficients 
of the first order, a function of three variables, three ; a 
function of four variables, four ; &c. 

It is now proposed to find the successive differentials 
of a function of two variables, and also the successive 
differential coefficients. 

We have already found 

, du 7 , du , 
au = —-dx J r— ay. 
dx dy 

(du du \ 

J x dx + Ty dy); 

and since, -7- and -j~ are functions of x and y, the 
dx dy y 

du du 
differentials -j-dx, -fdy, must each be differentiated 

with respect to both of the variables ; dx and dy being 

supposed constant : hence, 

/du \ d 2 u d 2 u 

d {di dx )=d^ dx + -d^r dxd * 

/du \ d 2 u d 2 u 

and d Xdy dy > = df V + ~dy& dyd * '' 

hence we have 

d 2 u d 2 u d u 

d 2 u = - 1 - 7 rdx 2 + 2^ r — r dxdy+ -r^dy 2 . 
dor dxdy dy 4 

If we differentiate again, we have 



88 ELEMENTS OF THE 

and consequently, 

d3 » = S^ 3 + 3 ^Sk d ^ +3 £k d ^ 2+ p^- 

It is very easy to find the subsequent differentials, by 
observing the analogy between the partial differentials and 
the terms of the development of a binomial. 

We also see that, a function of two variables has tivo 
partial differential coefficients of the first order, three of 
the second, four of the third, &c, 

88. There are several important results which may be 
deduced from the general development of the function of 
two variables (Art. 83). 

1st. If we make x = 0, and y = Q, u and each of 
the differential coefficients will become constant, and we 
shall have 



f^=u + \^A) 



dy 
1 nlH d 2 U d 2 U N 

+ &c., 

which is the development of any function of two variables 
in terms of their ascending powers, and coefficients which 
are dependent on the constants that enter the primitive 
function. 

2d. Jf, in the general development, we make y = 0, and 
k = 0, we shall have 



DIFFERENTIAL CALCULUS. 89 

7N du h d 2 u h 2 d 3 u h 3 , „ 

/( „ +A)=tt+ _ T+ __ + ____ + &c., 

which is the theorem of Taylor. 

3d. If we make y — 0, k = 0, and x = 0, we have 

„,. du h , d 2 u h 2 , d?u h? , « 

or, f(h) = 17+ W + U-~ + ^77^73+, &c. ; 
which is the theorum of Maclaurin. 

Implicit Functions. 

89. When the relation between a function and its 
variable is expressed by an equation of the form 

y = f( x ) 

in which y is entirely disengaged from x, y has been 
called an explicit, or expressed function of x (Art. 6). 
When y and x are connected together by an equation of 
the form 

/(*,y) - o, 

y has been called an implicit, or implied function of x 
(Art. 6.) 

It is plain, that in every equation of the form 

f(x,y) = 0, 

y must be a function of x, and x of y. For, if the 
equation were resolved with respect to either of them, the 
value found would be expressed in terms of the other 
variable and constant quantities. 



90 ELEMENTS OF THE 

90. If in the equation 

ur= f(x,y) = 0, 

we suppose the variables x and y to change their values 
in succession, any change either in x or y, will produce a 
change in u : hence, u is a function of x and y when 
they vary in succession. The value, however, which u 
assumes, when x or y varies, will reduce to when 
such a value is attributed to the other variable as will 
satisfy the equation 

f(x,y) = 0. 
We have from Art. 83, 

f( I 7 . 7 \ & U k 

J{x + h, y -f- k) — u = j— y-f- terms containing k 2 , 

du 

-T-h + terms containing h 2 , 

plus other terms containing kh, and the higher powers of 
h and k. 
But, since y is a function of x, we have 
k=Ph+P'h 2 , 
in which P is the differential coefficient of y regarded as 
a function of x. Substituting this value of k, and we 
have 

{x + h, y + k) — u = J-Ph + terms containing h 2 , 

du 

~fh + terms containing h 2 , 

plus other terms containing the higher powers of h. 



DIFFERENTIAL CALCULUS. 91 

But, in consequence of the relation between y and x, 
the first member of the equation will be constantly equal 
to 0. Hence, by the law of indeterminate coefficients 
(Alg., Art. 244), 

(du du\ 

du 

hence, P = ~- = — — . 

doc du 

dy 

Hence, the differential coefficient of y regarded as a 
function of x, is equal to the ratio of the partial differen- 
tial coefficients of u regarded as a function of x, and u 
regarded as a function ofy, taken with a contrary sign. 
Let us take, as an example, the equation 



then, 



hence, 

dy 
Although the differential coefficient of the first order is 
generally expressed in terms of x and y, yet y may be 
eliminated by means of the equation f(x,y) = 0, and the 
coefficient treated as a function of x alone. In the above 
equation we have 

y = VR* - x\ 



x > y) ■ 


= x 2 


+ 3/ 2 


-R 2 - 


-u — 


0; 


du 

dx~ 


2x, 


and 


du 
dy~ 


:Zy: 




du 












dx 
du 




X 

y ' 


dy^ 

dx 







92 ELEMENTS OF THE 



hence, 



dy _ x 

dx ~ VR 2 - x 2 



92. If it be required to find the second differential 
coefficient, we have merely to differentiate the first diffe- 
rential coefficient, regarded as a function of x, and divide 
the result by dx. Thus, if we designate the first diffe- 
rential coefficient by p, the second by q, the third by 
r, &c, we shall have 

dp do s 

ax ax 

93. To find the second differential coefficient in the 
equation of the circle, we have 

dy _ x 

y 



dx 



\dx) 



— ydx + xdy 
dy 



f 



— y + x 
i d 2 y dx 

hence, * = > 

dx 2 y 2 

and by substituting for -^ its value — — , we have 
dx y 

d 2 y_ a? + y 2 
dx 2 ~ y s 

1. Find the first differential coefficient of y, in the 
equation 

y % — 2mxy + x 2 — a 2 = u — 0, 

du , n du 

— = — 2?ny + 2x, -^- — 2y — 2mx: 



DIFFERENTIAL CALCULUS. 93 

, du r — 2 ///// + 2.r "1 ///■'/ — 07 

hence, -£ = — | 2_! — — 3. . 

dx L 2 // — 2 //W7 J y — mx 

2. Find the first differential coefficient of y in the 
equation 

if + 2xy + ^ — a 2 = 0. 

3. Find the first and second differential coefficients of y, 
in the equation 

y 3 — 3 «j?y + -r 3 = 0, 

— — 3 a; 2 — Say, —-~Sy 2 — Sax, 

dx dy 

hence, d -l = - **-*«y = ^ ~ ^ . 

dx Sy 1 —Sax y 2 — ax 

For the second differential coefficient, we have 

cPy y-acc){a d £-^)-{ay-a?)(2y d £-a)^ 
dx 2 (y 2 — ax) 2 

or, by substituting for -~- its value, and reducing, 
dx 

d?y _ _2xy i —6 ax 2 y 2 + 2yx* + 2 a 3 xy 
dx T ~ (y 2 -axf ' ' 

2xy (y 3 — 3 axy -f x 3 ) + 2a 3 xy t 
{y 2 -axf 

but from the given equation 

y 3 — Saxy 4-^ = 0. 

, d 2 y 2a 3 xy 

hence > tS - _ 7^> \3' 

dor (y— axy 



94 ELEMENTS OF THE 



Differential Equations of Curves. 

94. The Differential Calculus enables us to free an 
equation of its constants, and to find a new equation which 
shall only involve the variables and their differentials. 

If, for example, we take the equation of a straight line 

y — ax -f- by 
and differentiate it, we find 

dy _ 
dx 

and by differentiating again, 

The last equation is entirely independent of the values 
of a and b, and hence, is equally applicable to every 
straight line which can be drawn in the plane of the co- 
ordinate axes. It is called, the differential equation of 
lines of the first order. 

95. If we take the equation of the circle 

a? + y * = R\ 

and differentiate it, we find 

xdx + ydy = 0. 

This equation is independent of the value of the radius 
JR, and hence it belongs equally to every circle whose 
centre is at the origin of co-ordinates. 



DIFFERENTIAL CALCULUS. 



95 



96. If the origin of co-ordinates be taken in the circum- 
ference, the equation of the circle (An. Geom. Bk. Ill, 
Prop. I, Sch. 3) is 

from which we find 

X 

and by differentiating, 

_ x(2ydy -f- 2 xdx) — (y 2 + x^~)dx 

or by reducing 

(x 2 — y 2 ) dx + 2xydy = 0, 

which is the differential equation of the circle when the 
origin of co-ordinates is in the circumference. 

The last equation may be found in another manner. 

If we differentiate the equation of the circle, 

y 2 = 2Rx-x 2 i 

we have, after dividing by 2 

ydy = Rdx — xdx ; 

. ^ ydy + xdx 
hence, R — A • 

If this value of R be substituted in the equation of the 
circle, we have 

(x 2 — y 2 )dx + 2xydy = ; 
the same differential equation as found by the first method. 



96 ELEMENTS OF THE 

97. If we take the general equation of lines of the 
second order (An. Geom. Bk. VI. Prop. XII, Sch. 3), 

y 2 — mx 4- nx 2 , 

and differentiate it, we find 

2 ydy = mdx + 2 nxdx ; 

differentiating again, regarding dx as constant, we have, 
after dividing by 2, 

dy 2 + ycPy = ndx 2 . 

Eliminating m and n from the three equations, we obtain 

y'dx 2 -f- a?dy 2 — 2xydxdy-\- yx 2 d?y = 0, 

which is the general differential equation of lines of the 
second order. 

98. In order to free an equation of its constants, it will 
be necessary to differentiate it as many times as there are 
constants to be eliminated. For, two equations are neces- 
sary to eliminate a single constant, three to eliminate two 
constants, four to eliminate three constants, &c. : hence, 
one constant may be eliminated from the given equation 
and the first differential equation ; two from the given equa- 
tion and the first and second differential equations, &c. 

99. The differential equation which is obtained after the 
constants are eliminated, belongs to a species or order of 
lines, of which the given equation represents one of the 
species. 

Thus, the differential equation (Art. 94), 

da? 



DIFFERENTIAL CALCULUS. 97 

belongs to an order or species of lines of which the 
equation 

y = ax -+- b, 

represents a single one, for given values of a and b. 
The equation of a parabola is 

y 2 = 2px, 

and the differential equation of the species is 

2xdy — ydx — 0, or dy 2 + ycPy = 0. 

100. The differential equation of a species, expresses 
the law by which the variable co-ordinates change their 
values ; and this equation ought, therefore, to be indepen- 
dent of the constants which determine the magnitude, and 
not the nature of the curve. 

101. The terms of an equation may be freed from their 
exponents, by differentiating the equation and then com- 
bining the differential and given equations. 

Suppose, for example, 

P-=Q, 

P and Q being any functions of x and y. 
By differentiating, we obtain 

nP n ~ l dP=dQ: 

by multiplying both members by P, we have 

nP n dP = PdQ, 
and by substituting for P* its value, 

nQdP =. PdQ. 

7 



98 ELEMENTS OF THE 

The same result might also have been obtained by 
taking the logarithms of both members of the equation 



For, we have 
and (Art. 57). 



P n =Q. 
nlP = lQ, 



dP dQ 



hence, nQdP = PdQ. 



Of Vanishing Fractions, or those which take the 

r 

form — . 

J o 

102. It has been shown in (Alg. Art. Ill), that -, 
though a symbol of an undetermined quantity, may, under 
particular suppositions, become equal to 0, to a finite 
quantity, or to infinity. 

This symbol arises from the presence of a common 
factor in the numerator and denominator, which, becom- 
ing for a particular value of the variable, reduces the 

fraction to the form -. 


If we have, for example, a fraction of the form 

P(x - a) m 

Q(x - a) ni 

in which P and Q are functions of x, which do not re- 
duce to 0, for x = a, we have 

P(x - a) m 
Q{x - a) n "~ 0' 



DIFFERENTIAL CALCULUS. 99 

The value of this fraction will, however, be 0, finite or 
infinite, according as 

my n, m = n, m< n, 

for under these suppositions, respectively, it takes the form 

P( x-a) m - n P P 

Q ' Q' Q(x-a) n - m ' 

Let the numerator of the proposed fraction be desig- 
nated by X, and the denominator by X f , and let us sup- 
pose an arbitrary increment h to be given to x. The 
numerator and denominator will then become a function 
of x + h, and we shall have from the theorem of Taylor 

dX h 3 2 X 1? d 3 X h 3 „ 

A + ^T + "^ r lT2 + ^ r 1.2.3 + ' 

dXh_ d?X h 2 d 3 X h 3 
A + dx 1 + da* 1.2 + dx 3 1.2.3+ C ' 

If the value of x — a, reduces to the differential 
coefficients in the numerator as far as the rath order, and 
those of the denominator as far as the nxh order, the value 
of the fraction will become, 

d m X h m 



dx m 1.2.3.4 m 

~dFx h n 



+ &c. 



+ &c. 



dx 11 1.2. 3.4. ...n 

If we make h = 0, the value of the fraction will be- 
come 0, finite, or infinite according as 

m > n, m = n, m<n } 

and hence, if the value x = a, reduces to the same 
number of differential coefficients in the numerator and 



100 ELEMENTS OF THE 

denominator, the value of the fraction will be finite 
md equal to the ratio of the first differential coefficients 
which do not reduce to 0. 

103. Let us now illustrate this theory by examples. 
1. If in the fraction 

l-x n 

l-x' 

we make x — 1 , we have — . But 





dX_ n(xr _ x dX _ 

dx dx 


-l; 


in which, if we make x = 1, we have 




dX dX r 
— - — = — n, and — ; — = 
ax ax 


-i, 


dX 
hence, — = — 

dx -n, 

dX 


* 


dx • 





therefore, the value of the fraction when x =1, is +n. 
2. Find the value of the fraction 

ax 2 — 2 acx + ac 2 



bx 2 — 2 hex + be 2 ' 



when x — c, 



^ = 2ax-2ac r ^L = 2 bx-2bc, 

ax ax 

both of which become 0, when x = c. Differentiating 
again, we have 

-d^= 2a < ~aW^ h; 

hence, the true value of the fraction when x = c is -r-. 

o 



DIFFERENTIAL CALCULUS. 101 

3. Find the value of the fraction 

x 3 — ax 2 — a 2 x + a 3 , 

- - , when x — a. 

xr — a 1 

Ans. 0. 

4. Find the value of the fraction 

ax — x 2 , 

— — -— t = -, when x = a. 

a* — 2a s x + 2ax s — x* 

Ans. Qo. 

5. Find the value of 

a s -b l 



, when x = 0. 



x 

Ans. la — lb. 

6. What is the value of the fraction 

1 — sina? + cos-z , _._ 

, when 07 = 90°. 

sm# + cosa? — 1 

Ans. 1. 

7. What is the value of the fraction 

a — x — ala -\- alx , 

j- , when x = a. 

a— V2ax — x 2 

Ans. — 1. 

8. What is the value of the fraction 



-, when x=l. 



\ — x -\-lx 

Ans. — 2. 



9. What is the value of the fraction 

a n — x n , 

, when x = a. 



la — lx 

Ans. na r 



102 ELEMENTS OF THE 

104. It has been remarked (Art. 47), that the theorem 
of Taylor does not apply to the case in which a particular 
value attributed to x, renders any differential coefficient 
of the lunction infinite. Such functions are of the form 

{a*-a 2 ) m 
(x-af 9 

in which m and n are fractional. 

In functions of this form we substitute for x, a + h, 
which gives a second state of the function. We then 
divide the numerator and denominator by h raised to a 
power denoted by the smallest exponent of h, after which 
we make h = 0, and find the ratio of the terms of the 
fraction. 

When we place a + h for x, we have, in arranging 
according to the ascending powers of h % 

F(a + h) _ Alf + Bh b + Ch c + &c, 
F'(a + h)~ A f h a ' + Bh b ' + Ch* + &c. 

Now there are three cases, viz. : when 

a > a', a = a' y a < a 1 . 

In the first case the value of the fraction will be ; in 
the second, a finite quantity ; and in the third it w r ill be 
infinite. 

105, In substituting a + h for x y in the fraction 

(a* _ g?y 
{x — aY 



DIFFERENTIAL CALCULUS. 103 

. (2a h + h*)* fo , ,i 

we have i — ; — = (2a + h) i , 



and by making h = 0, which renders x = a the value of 
the fraction becomes 

(2a)\ 
2. Required the value of the fraction 

(x 2 -3ax+2a 2 Y 

j — - — when x = a. 

(x 3 - a 3 ) T 

Substituting a + h for x, we have 

1 — 11 

h 3 (-a + hy h 6 (-a + h) 3 

h* (3 a 2 + 3 ah + h 2 f (3 a 2 + 3 ah + A 2 ) 7 

which is equal to 0, when h—0. 

106. Remark. The last method of finding the value of 
a vanishing fraction, may frequently be employed advan- 
tageously, even when the value can be found by the 
theorem of Taylor. 

107. There are several forms of indetermination under 
which a function may appear, but they can all be reduced 

to the form — . 

1st. Suppose the numerator and denominator of the 

fraction 

X 

to become infinite by the supposition of x = a. The 
fraction can be placed under the form 



104 ELEMENTS OF THE 



_1_ 

X! 



1* 



which reduces to — , when X and X' are infinite. 

2d. We may have the product of two factors, one of 
which becomes and the other infinite, when a particular 
value is given to the variable. 

In the product PQ, let us suppose that x = a, makes 

P — and Q = cc . We would then write the product 

under the form, 

P 
PQ = - r 

~Q 

which becomes — when x = a. 


108. Let us take, as an example, the function 

(1 — a?)tangy«ra?; 

in which «• designates 180°. 

If we make x— 1, the first factor becomes 0, and the 
second infinite. But 

1 1 
tang — *x = j ; 

COt 7TX 

2 

hence, (1 — a?)tang— *•# = , 

cot — irx 
2 

2 
the value of which is — when x — 1 , 

•x 



DIFFERENTIAL CALCULUS. 105 



CHAPTER V. 

Of the Maxima and Minima of a Function of a 
Single Variable. 

109. If we have 

u = / (a?), 

the value of the function u may be changed in two ways ; 
first, by increasing the variable x ; and secondly, by dimin- 
ishing it. 

If we designate by v! the first value which u assumes 
when x is increased, and by u' f the first value which u 
assumes when x is diminished, we shall have three con- 
secutive values of the function 



Now, when u is greater than both v! and u f/ , u is said 
to be a maximum : and when u is less than both v! and 
u f/ , it is said to be a minimum. 

Hence, the maximum value of a variable function is 
greater than the value which immediately precedes, or the 
value that immediately follows : and the minimum value 
of a variable function is less than the value which imme- 
diately precedes or the value that immediately follows. 

110. Let us now determine the analytical conditions 
which characterize the maximum and minimum values of 
a variable function. 



106 ELEMENTS OF THE 

If in the function 

u = f(x), 

ihe variable x be first increased by h, and then diminished 
by A, we shall have (Art. 44), 

, Tf , 7N , du h , d 2 u h 2 , d?u h 3 « 

u' — f(x + h)=:u + - h -5-5 h-r^ h &c., 

y v ; da? 1 da? 2 1.2 da? 3 1.2.3 

„ -, 7N du h , d 2 u h 2 <Pu h 3 s 

u" = f(x — h) = u — r^ — h &c: 

J v ; dx 1 da* 1.2 da? 3 1.2.3 

and consequently, 

d 3 w /* 3 , - 

+ ^IX3 +&C '' 



?/ 


— u 


du h 

dx 1 


+ 


d?u 


h 2 




dx 2 


1.2 


— 


u = 


du h 
dx 1 


+ 


d?u 
dx 2 


h 2 
1.2 



d 3 u h 3 , . 



da? 3 1.2.3 

Now, if u has a maximum value, it will be greater 
than u? or w"; and hence, u' — u and w" — u will both 
be negative. If u is a minimum, it will be less than vf 
or w", and hence, u' — u and i*" — w will both be positive. 

Hence, in order that u may have a maximum or a 
minimum value, the signs of the two developments must 
be both minus or both plus. 

But since the terms involving the first power of h, in 
the two developments, have contrary signs, and since so 
small a value may be assigned to h as to make the first 
term in each development greater than the sum of all the 
other terms (Art. 44), it follows that u can have neither 
a maximum nor a minimum, unless 

dx 



DIFFERENTIAL CALCULUS. 107 

and the roots of this equation will give all the values of 
x which can render the function u either a maximum or 
a minimum. 

Having made the first differential coefficient equal to 0, 
the signs of the developments will depend on the sign of 
second differential coefficient. 

But since the signs of the first members of the equa- 
tions, and consequently of the developments, are both 
negative when u is a maximum, and both positive when u 
is a minimum, it follows that the second differential co- 
efficient will be negative when the function is a maximum, 
and positive when it is a minimum. Hence, the roots of 
the equation 7 

£ = o, 

ax 
being substitut2d in the second differential coefficient, will 
render it negative in case of a maximum, and positive in 
case of a minimum; and since there may be more than one 
value of x which will satisfy these conditions, it follows 
that there may be more than one maximum or one minimum. 
But if the roots of the equation 

du_ 

dx~ > 
reduce the second differential coefficient to 0, the signs 
of the developments will depend on the signs of the 
terms which involve the third differential coefficient ; and 
these signs being different, there can neither be a maxi- 
mum nor a minimum, unless the values of x also reduce 
the third differential coefficient to 0. When this is the 
case, substitute the roots of the equation 



I 08 ELEMENTS OF THE 

d / = o, 

ax 
in the fourth differential coefficient ; if it becomes negative 
there will be a maximum, if positive a minimum. If the 
values of x reduce the fourth differential coefficient to 0, 
the following differential coefficient must be examined. 
Hence, in order to find the values of x which w T ill render 
the proposed function a maximum or a minimum. 
1st. Find the roots of the equation 

£=0. 

ax 

2d. Substitute these roots in the succeeding differential 
coefficients, until one is found which does not reduce to 0. 
Then, if the differential coefficient so found, be of an odd 
order, the values of x ivill not render the function either 
a maximum or a minimum. But if it be of an even 
order, and negative, the function will be a maximum ; if 
positive, a minimum. 

111. Remark. Before applying the preceding rules to 
examples, it may be well to remark, that if a variable 
function is multiplied or divided by a constant quantity, 
the same values of the variable which render the function 
a maximum or a minimum, will also make the product or 
quotient a maximum or a minimum, and hence the con- 
stant will not affect the conditions of maximum or mini- 
mum. 

2. Any value of the variable which will render the 
function a maximum or a minimum, will also render any 
root or power a maximum or a minimum ; and hence, if 
a function is under a radical, the radical may be omitted. 



DIFFERENTIAL CALCULUS. 109 



EXAMPLES. 



1. To find the value of x which will render y a maxi- 
mum or a minimum in the equation of the circle 

y 2 + x 2 = R 2 , 

dy _ x 
dx~ y 1 

x 
making = 0, gives x = 0. 

The second differential coefficient is 

d?y _ x 2 4- y 2 
~dx T ~ p"-"' 

and since making x = 0, gives y = R, we have 

d 2 y 2_ 

dx 2 ~~ R 

which being negative, the value of x — renders y a 
maximum. 

2. Find the values of x which will render y a maximum 
or a minimum in the equation, 



y — a — bx + x 2 , 




differentiating, we find 




~r- = — b -f- 2a?, and 
aa? 





making, — 6 + 2a? = 0, gives # = — -; 

and since the second differential coefficient is positive, this 
value of x will render y a minimum. The minimum 



110 ELEMENTS OF THE 

value of y is found by substituting the value of x, in the 
primitive equation It is 

b 2 

y = a--. 

3. Find the value of x which will render the function 
u = a* + tfx — ctx 2 , 



a maximum or a minimum, 

— = b 3 — 2 c 2 x, hence x = — 

dx 2c 



and, S--** 

dor 

hence, the function is a maximum, and the maximum 
value is 

b 6 
u = a +7-5- 

4. Let us take the function 

u = 3 a 2 # 3 — 6 4 # + c 5 , 

du b 2 

we find -7- = 9 cPx 2 — 6 4 , and x = ± — 

aa? 3a 

The second differential coefficient is 

dhi 2 

3? = l8ax 

Substituting the plus root of x, we have 
(Pu , _ , 2 



DIFFERENTIAL CALCULUS. Ill 

which gives a minimum, and substituting the negative 
root, we have 

<Pu J2 

aB r = -e«B' > 

which gives a maximum. 

The minimum value of the function is, 

u = c ~FT ' 
9a 

and the maximum value 

u=c 5 +—. 
9a 

112. Remark. It frequently happens that the value 
of the first differential coefficient may be decomposed into 
two factors, X and X\ each containing x, and one of 
them, X for example, reducing to for that value of x, 
which renders the function a maximum or a minimum. 
When the differential coefficient of the first order takes 
this form, the general method of finding the second diffe- 
rential coefficient may be much simplified. For, if 

du 



-XX, 

ax 



we shall have 



d 2 u _ X'dX XdX 
dx 2 dx dx 



But by hypothesis X reduces to for that value of x 
which renders the function u a maximum or a minimum : 

, dhi X'dX 

hence, -j^. — ~ ~i — i 

dx 2 dx 



112 ELEMENTS OF THE 

from which we obtain the following rule for finding the 
second differential coefficient. 

Differentiate that factor of the first differential coef- 
ficient which reduces to 0, multiply it by the other factor, 
and divide the product by dx. 

5. To divide a quantity into two such parts that the mih 
power of one of the parts multiplied by the nth. power of 
the other shall be a maximum or a minimum. 

Designate the given quantity by a and one of the parts 
by x, then will a — x represent the other part. Let the 
product of their powers be designated by u ; we shall then 
have 

u — x m (a — x) n , 

du 
whence, y- = mx m ~ x (a - xf - nx m (a - xf~\ 
ax 

= (ma — mx — nx)x m ~ l (a — x) n ~\ 

and by placing each of the factors equal to 0, we have 

ma 

x= , x = 0, x = a. 

m + n 

The second differential coefficient corresponding to the 
first of these values, found by the method just explained, is 

g=-(« + n)«r- l (a-»)- 1 -; 

and substituting for x its value, it becomes 



(m + nY 



ifn-3 



hence, this value of x renders the product a maximum. 
The two other values of x satisfy the equation of the 



DIFFERENTIAL CALCULUS. 113 

problem, but do not satisfy the enunciation, since they are 
not parts o( the given quantity a. 

Remark. If in and n are each equal to unity, the quan- 
tity will be divided into equal parts. 

6. To determine the conditions which will render y a 
maximum or a minimum in the equation 

if — 2mxy + x 2 — a 2 = 0. 

The first differential coefficient is 

dy _ my — x m 
dx y — mx 9 

hence, my — x = 0, or y — — . 

J m 

Substituting this value of y in the given equation, we 
find 

ma 



x = 



VY 



and the value of y corresponding to this value of x is 

a 
Vl — m 2 

To determine whether y is a maximum or a minimum, 
let us pass to the second differential coefficient. We have 

-^ = (my-x)(y-mx)- 1 : 



(-2-0 



hence, il = _ 

dx 2 y — mx 



114 ELEMENTS OF THE 



and since ~ = 0, we have 
dx 



dx 2 y — mx 



and by substituting for y and x their values, we have 
d?y _ 1 



dx 2 aVl-m 2 ' 

hence, y is a maximum. 

7. To find the maximum rectangle which can be in- 
scribed in a given triangle. 

Let b denote the base of the triangle, h the altitude, 
y the base of the rectangle, and x the altitude. Then, 

u = xy = the area of the rectangle. 
But b : h : : y : h — x: 

bh — bx 



hence, 

and consequently, 



y = 



h 



bhx — bx 2 b , 7 „. 

== — (hx — or) , 



and omitting the constant factor, 

du , A 

—- = n — 2x, or # = — ; 
da? 2 ' 

hence, the altitude of the rectangle is equal to half the 
altitude of the triangle : and since 

d?u 



dx 2 
the area is a maximum. 



= -2, 



DIFFERENTIAL CALCULUS. 



115 



8. What is the altitude of a cylinder inscribed in a 
given cone, when the solidity of the cylinder is a maxi- 
mum ? 

Suppose the cylinder to be inscribed, 
as in the figure, and let 
AB = a, BC = b, AD = x, ED = y ; 
then, BD = a — x = altitude of the 
cylinder, and n y\a — a?) = solidity 
= v. (1) 

From the similar triangles AEB 
and ACB, we have 




: y : : a : b ; whence y = 



bx 



Substituting this value in equation (1), and we have 
,b 2 
v = ~lJ x \ a ~ x )' 



Omitting the constant factor — £-> we may write 

u = x 2 (a — .r) ; 
for the conditions which will make u a maximum will 
also make v a maximum (Art. 111). 
By differentiating, we have 



du 

-j- = 2(2-2? — Sx 2 , and — » = 2a — 6x. 

ax ax 



Placing 
we have 



ax 2 
2ax — Sx 2 = 0, 



x = 0, and x 



-a. 



Hence the altitude of the maximum cylinder is one-third 
the altitude of the cone. 



116* 



ELEMENTS OF THE 



Now, y / -ij=CF-PH= CD, 

y'- y CD 

— t— = prj = tangent of the angle CPD. 



and 



t> , • -, i CD PH 

But, by similar triangles ~pf)—~fjT • 

Now, the limiting ratio of the increment of the variable 
to that of the function, is that ratio which is independent 
of the value of h, and is obtained by making h equal to 
in the expression for the ratio of the increments (Art. 15.) 

It is evident that as h diminishes, the point C will ap- 
proach the point P, the point / will approach T, and the 
secant IC will approach the tangent TP ; and when h 
becomes equal to 0, the secant IC will coincide with the 
tangent TP. For every position of C we shall have 

CD PH 
-pjr~~^pfj= z tangent CPD = tangent CIH ; and when 

i r, CD PH nmrr dy 

C coincides with P, ~pfr = ~jvFj~ tangent PTH= -y- ; 

that is, the limiting ratio, or first differential coefficient, 
is equal to the tangent of the angle which the tangent 
line makes with the axis of abscissas. 

Of Tangents and Normals. 

114. Having found the value of 

dy 

dx 
we will now proceed to find the 
value of the subtangent, tangent, 
subnormal, and normal. T 




DIFFERENTIAL CALCULUS. 117 

We have (Trig. Th. II), 

1 : TR :: tangT : RP; 
that is, 1 : TR :: ^ : y 

hence, TR — y—- — sub-tangent. 

115. The tangent TP is equal to the square root of 
the sum of the squares of TR and RP ; hence, 



l~ dxJ 
TP = j/Y l+gp= tangent. 

116. From the similar triangles TPR, RPN, we have 

TR : PR : : PR : J?iV, 
hence, 2/ — : y : : y : .R7V, 

consequently, i?iV — y-±=z sub-normal. 

117. The normal PN is equal to the square root of the 
sum of the squares of PR and RN ; hence, 

PN = y\/l +%■= normal. 

v dor 

118. Let it be now required to apply these formulas to 
lines of the second order, of which the general equation 
(An. Geom. Bk. VI, Prop. XII, Sch. 3), is, 

y 2 = mx + nx 2 . 

Differentiating, we have 

dy _m-\-2nx _ m-\-2nx 
dx 2y 2Vmx + na?' 



118 



ELEMENTS OF THE 



substituting this value, we find 

, „„ dx 2{mx-\-nx 2 ) 

sub-tangent TK — y-~ — — -, 

y dy m + 2nx 



*Wl + £=V 



mx + nx 



sub-normal RN = y 



dy 



I mx + ?ix 2 ~\ 2 
\_m -\-2nx _\ 
m-\- 2nx 



dx 



PN = y\/l + ^=\/: 



mx + nx 2 + — (??i + 2nx) 2 . 



By attributing proper values to m and n, the above 
formulas will become applicable to each of the conic 
sections. In the case of the parabola, n = 0, and we have 



TR = 2x, 



RN=™, 
2 



TP= Vmx + lx 2 , 
PN = y mx + ±-r> 



119. It is often necessary to represent the tangent and 
normal lines by their equations. To determine these, in 
a general manner, it will be necessary first to consider the 
analytical conditions which render any two curves tangent 
to each other. 

Let the two curves, PDC, 
PEC, intersect each other at 
P and C. 

Designate the co-ordinates of 
the first curve by x and y, and 
the co-ordinates of the second by 
a/, y' . Then, for the common - 
point P, we shall have 

x = a/ t y = yf , 




DIFFERENTIAL CALCULUS. ] 19 

If we represent BG, the increment of the abscissa, by h, 
we shall have, from the theorem of Taylor (Ait. 44), 
^ ^ ^ -r. ^ ^ dy h dhj h' 1 d 3 y h 3 

da/ 1 dx 2 1 . 2 da/ 3 i .2.3 
hence, by placing the two members equal to each other, 
and, dividing by h, we have 

dy d 2 y h , _ dy' d 2 y' h t 

<fc + dz 2 1 .2 ^ ' do/ + da/ 2 1 .2 + 

If we now pass to the limit, by making h = 0, we shall 
have 

dy dy' 

dx ~"~ dxf ' 

in which case the point C will become consecutive with P, 
and the curve PEC tangent to the curve PDC. Hence, 
two lines will be tangent to each other, when they have 
a common point, and the first differential coefficient of 
the one equal to the first differential coefficient of the 
other, for this point. 

120. The equation of a straight line is of the form 
y = ax + b, 

dy 
nence, -^ = a. 

But the equation of a straight line passing through a 
given point, of which the co-ordinates are x", y r/ , is (An. 
Geom. Bk. II, Prop. IV), 

y—y' / = a[x — x"\ 



120 ELEMENTS OF THE 

But if the point whose co-ordinates are af\ y'\ is required 
to be on a given curve, these co-ordinates must satisfy 
the equation of that curve- If the straight line is required 
to be tangent to the curve at this particular point, the 

first differential coefficient -^, found from the equation 

dx 

of the curve, must take the particular value -£-; that is, 

we must have 

dy _ dy" 

dx dx // 
and the equation of the line tangent at the point whose 
co-ordinates are x // , y", will be 

121. Let it be required, for example, to make the line 
tangent to the circumference of a circle at a point of, 
which the co-ordinates are x", y" . 

The equation of the circle is x 2 + y 2 = R 2 ; 

dy x 
and, by differentiating, we have -=- = . 

But if the straight line is to be tangent to the circle, at 
the point whose co-ordinates are x", y'\ we must have 

dy" dy _ x x // 

'dx 77 = di~~~Y~~~¥ /; 
and by substituting this value in the equation of the line, 
and recollecting that x" 2 + y' n = R 2 , we have 

yy" -f- xx // = R 2 , 
which is the equation of a tangent line to a circle. 

122. A normal line is perpendicular to the tangent at 



DIFFERENTIAL CALCULUS. 121 

the point of contact, and since the equation of the tangent 
is of the form 

dy" , 

the equation of the normal, at the point whose co-ordin- 
ates are x", y" , will be of the form (An. Geom. Bk. II., 
Prop. VII., Sch. 2), 

dx" , 

y-y // = ~~d^\ x - <)» 

If we take the equation of any curve, and find the 

dx" 

value of for the particular point whose co-ordinates 

dy" 

are a/', y'\ and then substitute thac value in the above 
equation, we shall have the equation of the normal pas- 
sing through this point. 

The equation of the normal in the circle will take the form 

V" 
y=—x. 
y x' 

123. To find the equation of a tangent line to an ellipse 

at a point of which the co-ordinates are #", y", we have 

A 2 y //2 + -B 2 *" 2 = A 2 B 2 . 

By differentiating, we have 

dy^_ B 2 x" 

dx" ~~ A 2 y" ; 
hence, we have 

B 2 x // 

y-y"= - -fp-X x - rf'\ or JPyy" + &**' = A2 B 2 ; 

A 2 y" 
and for the normal y — y" = WZ7/C* ~ *")' 



122 



ELEMENTS OF THE 



124. To find the equation of a tangent to lines of the 
second order, of which the equation for a particular point 
(An. Geom. Bk. VI, Prop. XII, Sch. 3) is 

y" 2 — mx" + nx" 2 . 

By differentiating, we have 

dy" _ m-\-2 nx" 
~dx 7r ~ 2~f ' 

hence, the equation of the tangent to a line of the second 
order is 

m + 2 nx" 



y- y 2y" 

and the equation of the normal 

2y" 



y-y"= 



m -\-2nx 



{x-x"), 



jj(x-x"). 



Of Asymptotes of Curves. 



125. An asymptote of a curve is a line which continually 
approaches the curve, and becomes tangent to it at an 
infinite distance from the origin of co-ordinates. 

Let AX and AY be 
the co-ordinate axes, and 



?"=&--*"), 



dx 1 ' 

the equation of any tan 
gent line, as TP. 




DIFFERENTIAL CALCULUS. 123 

If in the equation of the tangent, we make in succes- 
sion y = 0, a?=0, we shall find 

If the curve CPB has an asymptote RE, it is plain 
that the tangent PT will approach the asymptote RE, 
when the point of contact P, is moved along the curve 
from the origin of co-ordinates, and T and D will also 
approach the points R and Y, and will coincide with 
them when the co-ordinates of the point, of tangency are 
infinite. 

In order, therefore, to determine if a curve have asymp- 
totes, we substitute in the values of AT and AD, the co- 
ordinates of the point which is at an infinite distance from 
the origin of co-ordinates. If either of the distances AT, 
AD, become finite, the curve will have an asymptote. 

If both the values are finite, the asymptote will be in- 
clined to both the co-ordinate axes : if one of the distances 
becomes finite and the other infinite, the asymptote will 
be parallel to one of the co-ordinate axes ; and if they both 
become 0, the asymptote will pass through the origin of 
co-ordinates. In the last case, we shall know but one 
point of the asymptote, but its direction may be deter- 
mined by finding the value of -j-, under the supposition 
that the co-ordinates are infinite. 

126. Let us now examine the equation 

y 2 — mx -f- nx 2 , 



124 ELEMENTS OF THE 

of lines of the second order, and see if these lines have 
asymptotes. We find 

AT = x - __-mx 



AD = y- 



m -{-2nx m + 2 nx 

mx -f 2 nx 2 mx 



2 y 2V??ix + nx 2 

which may be put under the forms 



AT= ~ m , AD 



'+2n Sy^ + n 



X 



X 



and making x = oo , we have 

AR — , and AE = 7= . 

2rc 2Vn 

If now we make n = 0, the curve becomes a parabola, 
and both the limits, AR, AE, become infinite : hence, 
the parabola has no rectilinear asymptote. 

If we make n negative, the curve becomes an ellipse, 
and AE becomes imaginary : hence, the ellipse has no 
asymptote. 

But if we make n positive, the equation becomes that 
of the hyperbola, and both the values, AR, AE, become 

finite. If we substitute for n its value —5, we shall have 

A 2 

AR=-A, and AE =+B. 



DIFFERENTIAL CALCULUS. 



125 



Differentials of the Arcs and Areas of Segments 
of Curves. 

127. It is plain, that the chord and arc of a curve will 
approach each other continually as the arc is diminished, 
and hence, we might conclude that the limit of their ratio 
is unity. But as several propositions depend on this rela- 
tion between the arc and chord, we shall demonstrate it 
rigorously. 

128. If we suppose the ordi- 
nate PR of the curve, POM to 
be a function of the abscissa, we 
shall have (Art. 19), 

PQ = h, and 

y'-y = MQ = {P + P'h)h; 

in which P = -%-. 

ax 



N 





11 



Hence, PM= Vh 2 +(P+P / h) 2 K 2 =h Vl + (P+P^) 2 . 
We also have NQ = Ph; 

hence, PN = Vh 2 + P 2 h 2 = hVl + P\ 

NM= NQ-MQ=- P'Ji: 
hence, we have 

PN + MN h Vl+P 2 - P'W VTTP* - P f h 



PM 



h Vl + (P + P f h) 2 Vl + (P + P'hf 



126 ELEMENTS OF THE 

of which the limit, by making h = 0, is 



= 1. 



Vi -f P 2 

But the arc POM can never be less than the chord PM, 
nor greater than the broken line PNM which contains it ; 
hence, the limit of the ratio 

POM 

PM ~ ' 

and consequently, the differential of the arc is equal to 
the differential of the chord. If we designate the arc by z, 
PM will be represented by z' — z, and we shall have 

z f ~z POM PM POM h , — — — 

-— —pm x -pq— pj V r x T Vl + ( p + p/A ) 2 ^ 

and, by passing to the limiting ratio, 



p*=^ * + % 



dz 



or dz = Vdx 2 + dy 2 ; 

that is, ^e differential of the arc of a curve, at any point, 
is equal to the square root of the sum of the squares of 
the differentials of the co-ordinates. 

129. To determine the differential of the arc of a circle 
of which the equation is 

X 2 + yl = R ^ 

xdx 
we have xdx + ydy = 0, or dy = ; 



. / x 2 dx 2 dx / 

whence, dz = v dx 2 + — ;r- = — Vx 2 -f- y 2 , 



V* -y ■ 



DIFFERENTIAL CALCULUS. 



127 



Rdx 



- ± 



Rdx 



Vw 



the same as determined in (Art 71). The plus sign is to 
be used when the abscissa x and the arc are increasing 
functions of each other, and the minus sign when they 
are decreasing functions (Art. 31). 

130. Let BCD be any segment 
of a curve, and let it be required 
to find the differential of its area. 

The two rectangles DCFE, 
DGME, having the same base 
DE, are to each other as DC to 
EM; and hence, the limit of their 
ratio is equal to the limit of the ratio of DC to EM, 
which is equal to unity. 

But the curvelinear area DCME is less than the rect- 
angle DGME, and greater than the rectangle DCFE : 
hence, the limit of its ratio to either of them will be 
unity. But, 





G 


At 




>- 


F 


B 

! 


' 







]) 



DCME 
DE 



DCME DEFC 

x 



DE 



DEFC 



DCx 



DCME 
DEFC 



or by representing the area of the segment by s and the 
ordinate DC by y, and passing to the limit, we have 



ds 
dx 



y> 



or ds = ydx ; 



hence, the differential of the area of a segment of any 
curve, is equal to the ordinate into the differential of the 
abscissa. 



128 ELEMENTS OF THE 

131. To find the differential of the area of a circular 
segment, we have 

x 2 + y 2 = R 2 , and y= VR 2 -a?; 

hence, ds = dx vR z — x 2 . 

The differential of the segment of an ellipse, is 

ds — — dx^/A 2 — x 2 , 
A. 

and of the segment of a parabola 

ds — dxV2px. 

Signification of the Differential Coefficients. 



132. It has already been shown that, if the ordinate of 
a curve be regarded as a function of the abscissa, the first 
differential coefficient will be equal to the tangent of the 
angle which the tangent line forms with the axis of abscis- 
sas (Art. 113). We now propose to show the signification 
of the second differential coefficient, the ordinate being re- 
garded as a function of the abscissa. 

Let AP be the abscissa 
and PM the ordinate of a 
curve. From P lay off 
on the axis of abscissas 
PP f = h, and PP f/ = 2h. 
Draw the ordinates PM, 
P'M,P"M'; also the lines 
MMN, MM" ; and lastly, 
MQ, M'Q f , parallel to the 




DIFFERENTIAL CALCULUS. 



129 



axis of abscissas. Then will M'Q = NQ', and we shall 
have 

PM=y, 

dx 1 dx 2 1.2 



Ws= y + £A + 4£JL 



""-'+2T+-&TF+*- 



J p^f // -P / AT=JI // Q / =^^+ f y 9 3A2 -f- &c. 
dx dx 2 1.2 

M>'Q'- M'Q == + M"N = -!%h 2 + &c. 

aar 

Now, since the sign of the first member of the equation 
is essentially positive, the sign of the second member will 
also be positive (Alg. Art. 85). But by diminishing h, the 
sign of the second member will depend on that of the 
second differential coefficient (Art. 44) : hence, the second 
differential coefficient is positive. 



If the curve is below 
the axis of abscissas, 
the ordinate s will be 
negative, and it is easily 
seen that we shall then 
have 



MtiQ!—M'Q= — MW=-^W + &c. 





P P' 


P" 


A 






Q 










M 


Q' 


















N 












M" 



130 



ELEMENTS OF THE 



Now, since the first member is negative, the second 
member will b*e negative : hence we conclude that, if a 
curve is convex towards the axis of abscissas, the ordi- 
nate and second differential coefficient will have like signs. 



133. Let us now con- 
sider the curve CMM'M", 
which is concave towards 
the axis of abscissas. We 
shall have, 




PM = y, 



y dx 1 dot? 1.2 

y doc 1 T da? 1 . 2 

P'M- PM=M'Q = ^A + J*V *L + & 

dx 1 dor 1.2 

P"M"-PM f = M"Q' = d ith+ f% Sh2 + &c., 
do? aor 1 . 2 

M"Q'- M'Q = - NM"= -^X-h 2 + &c. 

s 
j 

But since the first .member of the equation is negative, 

the essential sign of the second member will also be 
negative : hence, the second differential coefficient will 
be negative. 



DIFFERENTIAL CALCULUS. 



131 



If the curve is below the 
axis of abscissas, the ordi- 
nate will be negative, and it 
is easily seen that we should 
then have 



p p 


P 


a 


M 




Q 








Q' 

W 7 ' 




M> 





N 



M"Q- M'Q = + NM"= ^Ltf + &c. ; 

dot? 

hence we conclude that, if a curve is concave towards the 
axis of abscissas, the ordinate and second differential 
coefficient will have contrary signs. 

The ordinate will be considered as positive, unless the 
contrary is mentioned. 

134. Remark 1. The co-ordinates x and y, determine 
a single point in a curve, as M. The differential of y is 
derived from the ordinate PM, and is what QM' becomes 
when the ordinates P'M and PM become consecutive. 

The second differential of y is derived from M'Q, in 
the same way that dy is derived from the primitive func- 
tion y. It is, indeed, what M'Q' becomes, when M"Q' 
becomes consecutive with MQ. The abscissa x being 
supposed to increase uniformly, the difference between 
PP' and P / P // is : and therefore the second differential 
of x is 0. The co-ordinates x and y, and the first and 
second differentials determine three points, M, M' 9 M\ 
consecutive with each other. 

135. Remark 2. When the curve is convex towards 



132 ELEMENTS OF THE 

the axis of abscissa, the first differential coefficient, which 
represents the tangent of the angle formed by the tangent 
line with the axis of abscissas, is an increasing function 
of the abscissa : hence, its differential coefficient, that is, 
the second differential coefficient of the function, ought 
to be positive (Art. 31). 

When the curve is concave, the first differential coeffi- 
cient is a decreasing function of the abscissa ; hence, the 
second differential coefficient should be negative (Art. 31). 

Examination of the Singular Points of Curves. 

136. A singular point of a curve is one which is dis- 
tinguished by some particular property not enjoyed by 
the points of the curve in general. 

Let us, as a first example, find the points of a curve, 
through which the tangent lines will be parallel or per- 
pendicular to the axis of abscissa* 

137. Since the first differential coefficient expresses the 
value of the tangent of the angle which the tangent line 
forms with the axis of abscissas, and since the tangent is 
0, when the angle is 0, and infinite when the angle is 90°, 
it follows that the roots of the equation 

ax 

will give the abscissas of all the points at which the tan- 
gent is parallel to the axis of abscissas, and the roots of 
the equation 

dy dx _ 

£ = <*' ° r Ty = °' 



DIFFERENTIAL CALCULUS. 133 

will give the abscissas of all the points at which the tan- 
gent is perpendicular to the axis of abscissas. 

138. If a curve from being convex towards the axis of 
abscissas becomes concave, or from being concave becomes 
convex, the point at which the change of curvature takes 
place is called a point of inflexion. 

Since the ordinate and differential coefficient of the 
second order have the same sign when the curve is convex 
towards the axis of abscissas, and contrary signs when it 
is concave, it follows that at the point of inflexion, the 
second differential coefficient will change its sign. There- 
fore between the positive and negative values there will be 
one value of x which will reduce the second differential 
coefficient to or infinity (Alg. Art. 310) : hence the roots 
of the equations 





d?y d?y 




will give the abscissas of the points of inflexion. 




139. Let 
the equation 


us now apply these principles in 
of the circle 

x 2 + y 2 = R 2 . 


discussing 


We have, 


by differentiating, 




and placing 


dy x 
dx~ y' 

= 0, we have x — 0. 

y 




Substituting this value in the equation of the 

have 

y=±R; 


curve, we 



134 ELEMENTS OF THE 

hence, the tangent is parallel to the axis of abscissas at 
the two points where the axis of ordinates intersects the 
circumference. 
If we make 





dy = 
dx 


— 


X 
— =00, 

y 


or 


_y_ = 

X 


:0, 




we 


have y = 





; substituting 


this value in 


the 


equation, 


we 


find 




x= ± 


R, 









and hence, the tangent is perpendicular to the axis of 
abscissas at the points where the axis intersects the cir- 
cumference. 

The second differential coefficient is equal to 

HL 

y 3 ' 

which will be negative when y is positive, and positive 
when y is negative. Hence, the circumference of the 
circle is concave towards the axis of abscissas. 

If we apply a similar analysis to the equation of the 
ellipse, we shall find the tangents parallel to the axis of 
abscissas at the extremities of one axis, and perpendicular 
to it at the extremities of the other, and the curve concave 
towards its axes. 

140. Let us now discuss a class of curves, which may 
be represented by the equation 

y = b ± c(x — a) m , 

in which we suppose c to be positive or negative, and 
different values to be attributed to the exponent m. 



DIFFERENTIAL CALCULUS. 



135 



1st. When c is positive, and m entire and even 



By differentiating, we have 
dy 



, — vicix — a) , 
dx 

^ = m{m-l)c{x-a) m -\ 



If we place the value -+- = 0, we find x = a, and sub- 
dx 

stituting this value in the equation of the curve, we find 

y — b : 

hence, x = a, y = b, are the co-ordinates of the point 
at which the tangent line is parallel to the axis of 
abscissas. 

Since m is even, m — 2 will 
also be even, and hence the second 
differential coefficient will be posi- 
tive for all values of x. The curve 
will therefore be convex towards 
the axis of X, and there will be 
no point of inflexion. 

The value of x = a renders the ordinate y a minimum, 
since after m differentiations a differential coefficient of an 
even order becomes constant and positive (Art. 110). 

The curve does not intersect the axis of X, but cuts the 
axis of Y at a distance from the origin expressed by 




y = b + ca 1 



136 



ELEMENTS OF THE 



141. 2d. When c is negative, and m entire and even. 



We shall have, by differentiating, y = b — c(x—a) v 



dy _ 



dx 



= — mc (x — a J 



and 



^l = - m ( m -.l)c(x-a) m -\ 



The discussion is the same as 
before, excepting that the second 
differential coefficient being nega- 
tive for all values of x, the curve 
is concave towards the axis of 
abscissas, and the value of x = a, — 
renders the ordinate y a maxi- 
mum (Art. 110). 

142. 3d. When c is plus or minus, and m entire and 
uneven. 

We shall have, by differentiating, 



mc(x — a) m -\ 



and 



dy__ 
dx 

^ 2 =±m(m-l)c(x-a) m -\ 



The first differential coefficient will be 0, when x = a ; 
hence, the tangent will be parallel to the axis of abscissas, 
at the point of which the co-ordinates are x = a, y = b. 



DIFFERENTIAL CALCULUS. 



137 



Since the exponent m — 2 is 
uneven, the factor (x — «) m ~~ 2 will 
be negative when x < a, and 
positive when x > a; hence, this 
factor changes its sign at the 

point of the curve of which the 

abscissa is x = a. 

If c is positive, the second differential coefficient will be 
negative for #<«, and positive for x>a: hence there will 
be an inflexion when x — a. If c were negative, the curve 
would be first convex and then concave towards the axis 
of abscissas, but there would still be an inflexion at the 
point x = a. At this point the tangent line separates the 
two branches of the curve. 

There will, in this case, be neither a maximum nor a 
minimum, since after m differentiations a differential coef- 
ficient of an odd order, will become equal to a constant 
quantity (Art. 110). 

143. 4th. When c is positive or negative, and m a 
fraction having an even numerator, as m = — . 

o 

By differentiating, and supposing c positive, we have 

2c 



dy 2 . 
-r- = -^c(x 
dx 3 v 



I- 



S(x-af 



d?l 

dx 2 



2c 



9(x-af 



If we make x = a, the first differential coefficient will 
become infinite ; and the tangent will be perpendicular to 



138 



ELEMENTS OF THE 




the axis of abscissas, at the point of which the co-ordinates 
are x = a, y = b. 

In regard to the second differen- 
tial coefficient, it will become infi- 
nite for x = a, and negative for 
every other value of x, since the 
factor (x — a) of the denominator 
is raised to a power denoted by an 
even exponent. Hence, the curve 
will be concave towards the axis of 
abscissas. 

If we take the equation of the curve 

y z= b + c (x — a) 3 , 

and make x = a + h, and x = a — h, we shall have, in 
either case, 

y — b + ch 3 ; 

and hence, y will be less for x = a, than for any other 
value of x, either greater or less than a. Hence, the 
value x = a, renders y a minimum. 

If c were negative, the equation would be of the form 







y = b 


— c{x — a) 3 ; 


and we 


should h 


ave, by differentiating, 






dy 
dx 


2c 
3{x-a.y 


and 




d?y 


2c 


dx 2 n 


4 ' 



9(a?-a) 3 



DIFFERENTIAL CALCULUS. 



139 



The first and second differen- 
tial coefficients will be infinite for 
X — a, and the second differential 
coefficient will be positive for all 
values of x greater or less than a; 
and hence, the curve will be con- 
vex towards the axis of abscissas. 

If, in the equation of the curve 




y = 6 — c(x — ay, 



we make x — a + h, and x 
either case, 

y = b- 



a — h, we shall have, in 



,2 

ch*; 



and hence, y will be greater for x = a, than for any other 
value of x either greater or less than a. Hence, the 
value x = «, renders y a maximum. 

144. Remark. The conditions of a maximum or a 
minimum deduced in Art. 110, were established by means 
of the theorem of Taylor. Now, the case in which the 
function changes its form by a particular value attri- 
buted to x, was excluded in the demonstration of that 
theorem (Art. 45). Hence, the conditions of minimum 
and maximum deduced in the two last cases, ought 
not to have appeared among the general conditions of 
Art. 110. 

We therefore see that there are two species of maxima 
and minima, the one characterized by 



f=°> 



the other by 



dy 
dx 



= oo . 



140 ELEMENTS OF THE 

In the first, we determine whether the function is a 
maximum or a minimum by examining the subsequent 
differential coefficient ; and in the second, by examining 
the value of the function before and after that value of x 
which renders the first differential coefficient infinite. 

The branches DE, ME, which are both represented by 
the equation. 

2 

y = b ± c(x — a) 7 , 

are not considered as parts of a continuous curve. For, 
the general relations between y and x which determine 
each of the parts DE, ME, is entirely broken at the 
point M, where x = a. The two parts are therefore 
regarded as separate branches which unite at M. The 
point of union is called a cusp, or a cusp point. 

145. 5th. When c is positive or negative and m a 

3 
fraction having an even denominator, as m =-r- 

Under this supposition the equation of the curve will 
become 







y = b 


± c(x 


-a)\ 


and 


by 


differentiating, we 


have 








dy 
dx 


-j- 


3c 




— i» 
4 (x — a) * 


find 




&y 
dx 2 '' 


— -T- 


3c 




— -T- 

4. 


5 * 
4(07 — fl)< 



DIFFERENTIAL CALCULUS. 



141 



U 



The curve represented by this 
equation will have two branches : 
the one corresponding to the plus 
sign will be concave towards the 
axis of abscissas, and the one cor- 
responding to the minus sign w r ill be 
convex. Every value of x less than 
a will render y imaginary. The co-ordinates of the point 
M, are x = a, y = b. 

146. 6th. When c is positive or negative and m a 
fraction having an uneven numerator and an uneven de- 
nominator, as m= — . 
5 

Under this supposition the equation will become 



y = b ± c(x 
and by differentiating, we have 



dx 

o?y _ 
dx 2 ~^ 



3c 



5(x — a) 5 
3.2c 



1 ' 
a) 5 



5.5(a? 

from which we see that if we use the superior sign of the 
first equation, the curve will be convex towards the axis 
of abscissas for x < a, that there will be a point of inflexion 
for x = a, and that the curve will be concave for x > a. 
If the lower sign be employed, the first branch will become 
concave, and the other convex. 

147. The cusps, which have been considered, were 
formed by the union of two curves that were convex to- 



142 



ELEMENTS OF THE 



wards each other, and such are called, cusps of the first 
order. 

It frequently happens, however, that the curves which 
unite, embrace each other. The equation 

furnishes an example of this kind. By extracting the 
square root of both members and transposing, we have 



y = x 2 ±x' 



y _ 



dx 2 



2=fc 



2 ' 2 



■X' 



and by differentiating 

dx 2 

We see by examining 
the equations, that the curve 
has two branches, both of 
which pass through the 
origin of co-ordinates. The 
upper branch, which corres- 
ponds to the plus sign, is constantly convex towards the 
axis of abscissas, while the lower branch is convex for 




x< 



64 



and concave for x > 



64 



and x < 1 . At 



225 225 

the last point the curve passes below the axis of abscissas 

and becomes convex towards it. If we make the first dif- 
ferential coefficient equal to 0, we shall find x — 0, and 
substituting this value in the equation of the curve, gives 
y = ; and hence, the axis of abscissas is tangent to both 
branches of the curve at the origin of co-ordinates. At 
this point the differential coefficient of the second order 
is positive for both branches of the curve, hence they 



DIFFERENTIAL CALCULUS. 



143 



arc both convex towards the axis. When the cusp is 
formed by the union of two curves which, at the point 
of contact, lie on the same side of the common tangent, it 
is called a cusp of the second order. 

148. Let us, as another example, discuss the curve 
whose equation is 

V 



b ±(x — a) yx — c. 
By differentiating, we obtain 

y*x — c 



dy 
dx 



x — a 



2V~c 




We see, from the equa- 
tion of the curve, that y will 
be imaginary for all values 
of x less than c. 

For x=c, we have y — b; 
and for x > c, we have two 
values of y and conse- 
quently two branches of 
the curve, until x = a when they unite at the point M. 
For x > a there will be two real values of y and conse- 
quently two branches of the curve. The point M, at 
which the branches intersect each other, is called a mul- 
tiple point, and differs from a cusp by being a point 
of intersection instead of a point of tangency. At the 
multiple point M there are two tangents, one to each 
branch of the curve. The one makes an angle with the 
axis of abscissas, whose tangent is 



+ V^ 37 



10 



144 ELEMENTS OF THE 

the other, an angle whose tangent is 



— y/a — c* 

149. Besides the cusps and multiple points which have 
already been discussed, there are sometimes other points 
lying entirely without the curve, and having no connexion 
with it, excepting that their co-ordinates will satisfy the 
equation of the curve. 

For example, the equation 

ay 2 — x 3 -f bx 2 = 0, 

will be satisfied for the values 
a? = ± 0, y = ±0 ; and hence, 
the origin of co-ordinates A, 
satisfies the equation of the 
curve, and enjoys the property 
of a multiple point, since it is 
the point of union of two values 
of x, and two values of y. 

If we resolve the equation with respect to y, we find 



*v^ 



and hence, y will be imaginary for all negative values of 
a?, and for all positive values between the limits x = and 
x = b. For all positive values of x greater than b } the 
values of y will be real. 

The first differential coefficient is 

dy _ x(3x—2b) 
d* ~ 2 Vax?(x - b) ' 



DIFFERENTIAL CALCULUS. 145 

or by dividing by the common factor x, 

dy _ Sx — 2b 
dx~ 2 Va(x-b) 

and making x = 0, there results 

dy _ 2b 

doc 2 V-ab ' 

which is imaginary, as it should be, since there is no point 
of the curve which is consecutive with the isolated or con- 
jugate point. The differential coefficients of the higher 
orders are also imaginary at the conjugate points. 

150. We may draw the following conclusions from the 
preceding discussion. 

1st. The equation -^- = 0, determines the points at 
which the tangents are parallel to the axis of abscissas. 

2d. The equation -y- — oo , determines the points of 
ax 

the curve at which the tangents are perpendicular to the 

axis of abscissas. The two last equations also determine 

the cusps, if there are any, in all cases where the 

tangent at the cusps is parallel or perpendicular to the 

axis of abscissas. 

d?y cPy 

3d. The equation -^ = 0, or — -jJ-=ao determines 

the points of inflexion. 

4th. The equation — ^ = an imaginary constant, in 
dicates a conjugate point. 



146 



ELEMENTS OF THE 



CHAPTER VII. 



Of Oscillatory Curves — Of Evolutes. 



151. Let PT be tangent to the curve ABP at the point 
P, and PN a normal at the same point : then will PT 
be tangent to the circumference of every circle passing 
through P, and having its centre in the normal PN. 

It is plain that the cen- 
tre of a circle may be 
taken at some point C, 
so near to P, that the cir- 
cumference shall fall with- 
in the curve APB, and 
then every circumference 
described with a less ra- 
dius, will fall entirely 
within the curve. It is 

also apparent, that the centre may be taken at some point 
C, so remote from P, that the circumference shall fall 
between the curve APB and the tangent PT t and then 
every circumference described with a greater radius will 
fall without the curve. Hence, there are two classes of 
tangent circles which may be described; the one lying 
within the curve, and the other without it. 




DIFFERENTIAL CALCULUS. 



147 




ACE 



152. Let there be 
three curves, APB, 
CPD, EPF, which 
have a common tan- 
gent TP, and a com- 
mon normal PN ; then 
will they be tangent to 
each other at the point 
P. It does not follow, 
however, from this cir- 
cumstance, that each curve will have an equal tendency to 
coincide with the tangent TP, nor does it follow that any 
two of the curves CPD, EPF, will have an equal ten- 
dency to coincide with the first curve APB. 

It is now proposed to establish the analytical 
conditions which determine the tendency of curves to 
coincide with each other, or with a common tangent. 

Designate the co-ordinates of the first curve APB by 
x and y, the co-ordinates of the second CPD by a/, y' , 
and the co-ordinates of the third EPF by x" , y" . If we 
designate the common ordinate PR by y, y f , y" , we shall 
then have 

h 3 



7 y_r dx 1 ^^1.2 dx 3 1.2.3 



+ &c, 



„. , dy' h , dV h 2 , d 3 y / h 3 , \ 



p/ . „_, dy" h , $fW d?y" h 3 
nR f = y" + -^; — + - V JL_ + 



da/' 1 ^ /2 1.2 ' da/ /3 1.2.3 



4- &c. 



But since the curves are tangent to each other at the 
point P, we have (Art. 119), 



148 ELEMENTS OF THE 

y = y> = y», and % = % = %>•■ hence, 

"-^"W dWVl.8 + W rfWl.2.3 + " 
„ /dfy dV'\ /j 2 , fd?y d?y"\ h 3 . 

Now, in order that the first curve APB shall approach 
more nearly to the second CPD than to the third EPF> 
we must have 

d<d', 

and consequently, 

h 2 h 3 1? h 3 

A— + B-?— + &c.,<A / -^-+ J B / — — + &c, 
1.2 1.2.3 ' 1.3 1.2.3 

in which we have represented the coefficients in the first 
series by A, J5, C, &c., and the coefficients in the second 
by A', J5', C, &c. 

Now, the limit of the first member of the inequality will 
always be less than the limit of the second, when its first 
term involves a higher power of h than the first term of 
the second. For, if A = 0, the first member will involve 
the highest power of h, and we shall have 

A 3 h 2 h 3 

B-t- + &c, < A'— + B'-^— + &c, 
1.2.3 1.2 1.2.3 



and by dividing by h t 

B-^— + &c.,<A' — ±B'— — 
1.2.3 1.2 J 1.2.3 



and by passing to the limit 



DIFFERENTIAL CALCULUS. 149 

But when A = 0, we have 

o?y _ tfy' 
da? ~ do/ 2 ' 

and hence, when three curves have a common ordinate, the 
first will approach nearer to the second than to the third, 
if the number of equal differential coefficients between the 
first and second is greater than that between the first and 
third. And consequently, if the first and second curves 
have m + 1 differential coefficients which are equal to 
each other, and the first and third curves only m equal dif- 
rential coefficients, the first curve will approach more 
nearly to the second than to the third. Hence it appears, 
that the order of contact of two curves will depend on 
the number of corresponding differential coefficients which 
are equal to each other. 

The contact which results from an equality between the 
co-ordinates and the first differential coefficients, is called 
a contact of the first order, or a simple tangency (Art. 119). 
If the second differential coefficients are also equal to each 
other, it is called a contact of the second order. If the first 
three differential coefficients are respectively equal to each 
other, it is a contact of the third order; and if there are m 
differential coefficients respectively equal to each other, it 
is a contact of the mth order. 

153. Let us now suppose that the second line is only 
given in species, and that values may be attributed at 
pleasure to the constants which enter its equation. We 



150 ELEMENTS OF THE 

shall then be able to establish between the first and second 
lines as many conditions as there are constants in the 
equation of the second line. Jf, for example, the equation 
of the second line contains two constants, two conditions 
can be established, viz. : an equality between the co- 
ordinates, and an equality between the first differential 
coefficients ; this will give a contact of the first order. 

If the equation of the second curve contains three con- 
stants, three conditions may be established, viz. : an equality 
between the co-ordinates, and an equality between the first 
and second differential coefficients. This will give a con- 
tact of the second order. If there are four constants, we 
can obtain a contact of the third order ; and if there are 
#i + l constants, a contact of the mth order. 

It is plain, that in each of the foregoing cases the highest 
order of contact is determined. 

The line which has a higher order of contact with a 
given curve than can be found for any other line of the 
same species, is called an osculatrix. 

Let it be required, for example, to find a straight line 
which shall be osculatory to a curve, at a given point of 
which the co-ordinates are x" > y" '. 

The equation of the right line is of the form 

y = ax + b, 

and it is required to find such values for the constants a 
and b as to cause the line to fulfil the conditions, 

* = *", y = y", and | = g. 



DIFFERENTIAL CALCULUS. 151 

By differentiating the equation of the line, we have 

dy 

— = a : 

dx ' 

and since the line passes through the point of osculation 
y-y"= d JL{ x -,/'). 

Substituting for -j- its value -777 > we nave 

for the equation of the osculatrix. 
In the equation of the circle 

x*+y 2 = R 2 , 

6 a dy _ x _ dy" _ xf f 
dx" y~~ dx" ~ y" 

hence, the equation of the osculatrix of the first order, to 
the circle, is 

x" 

y -y"=---(x-x"\ 

or by reducing yy" + xx" = R 2 . 

154. If » and /3 represent the co-ordinates of the centre 
of a circle, its equation will be of the form 

(x-«Y + (y-pY = R 2 . 

If this equation be twice differentiated, we shall have, 

(x — a)dx-\-(y — f)dy = 0, 

dx 2 + dy 2 + (y-p)d 2 y = 0; 



352 



ELEMENTS OF THE 



and by combining the three equations, we obtain, 

dx 2 + dy* 



y — P 



d 2 y 



)■ 



dx \ d 2 y 
dxd z y 



If it be now required to make this circle osculatory to 
a given curve, at a point of which the co-ordinates are a/', 
y ,f , we have only to substitute in the three last equations, 
the values of 



dy _ dy" 
dx'dx 77 ' 



d?y _ (Py^ 
'dx 2 ~lx 7j2 



deduced from the equation of the curve, and to suppose, at 
the same time, the co-ordinates x and y in the curve to 
become equal to those of x and y in the circle. 

If we suppose x", y" to become general co-ordinates 
of the curve, the circle will move around the curve, con- 
tinually changing its radius, and will become osculatory 
at all the points in succession. 

155. If the circle CD 
be osculatory to the curve 
EF, at the point P, we 
shall have 

h 3 
qs=Cx + YY% + &c '> 

for h positive ; and 

h 3 
1.2.3 



q V = C X 



+ &c, 




DIFFERENTIAL CALCULUS. 153 

for h negative: hence, the two lines qs, q f s f , have contrary 
signs. The curve, therefore, lies above the oscillatory cir- 
cle on one side of the point P, and below it on the other, 
and consequently, divides the oscillatory circle at the point 
of osculation. Hence, also, the oscillatory circle separates 
the tangent circles which lie without the curve from those 
which lie within it (Art. 151). 

In every osculatrix of an even order the first term in the 
values of qs, q's', will, in general, contain an uneven power 
of h ; and hence their signs may be made to depend on 
that of h. The curve will therefore lie above the oscu- 
latrix on one side of the point P, and below it on the 
other ; and hence, every osculatrix of an even order, will 
in general be divided by the curve at the point of oscula- 
tion. 

156. The first differential equation of Article 154, 

(x — et)dx -\-(y— $)dy = 
may be placed under the form 

dx. 

It we make the circle osculatory to the curve we have 

x =r x // , y = y" , and 
dx dx' f . 

_ ; hence , 

dx" 

which is the equation of a normal at the point whose co- 
ordinates are x" y" (Art. 122). But this normal passes 
through the point whose co-ordinates are * and /3. Hence, 
the normal drawn through the point of osculation, will 
contain the centre of the osculatory circle. 

157. It was shown in (Art. 155) that the osculatory cir- 
cle is, in general, divided by the curve at the point of oscu- 



154 ELEMENTS OF THE 

lation. The position of the curves with respect to each 
other indicates this result. 

For, the osculatory circle is always symmetrical with 
respect to the normal, while the curve is, in general, not 
symmetrical with respect to this line. If, however, the 
curve is symmetrical with respect to the normal, as is the 
case in lines of the second order when the normal coincides 
with an axis, the curve will not divide the osculatory circle 
at the point of osculation ; and the condition which renders 
the second differential coefficients in the curve and circle 
equal to each other, will also render the third differential 
coefficients equal, and the contact will then be of the third 
order. 

158. The radius of the osculatory circle 

R=± (d^±dfl 
doccPy 

is affected with the sign plus or minus, and it may be well 
to determine the circumstances under which each sign is 
to be used. 

If we suppose the ordinate to be positive, we shall have 
(Art. 133) 

cPy 

•j?-, and consequently <Fy 

CLOU 

negative when the curve is concave towards the axis ot 
abscissas, and positive when it is convex. If then, we 
wish the radius of the osculatory circle to be positive for 
curves which are concave towards the axis of abscissas, we 
must employ the minus sign, in which case the radius will 
be negative for curves which are convex. 



DIFFERENTIAL CALCULUS. 155 

159. If the circumferences of two circles be described 
with different, radii, and a tangent line be drawn to each, it 
is plain that the circumference which has the less radius 
will depart more rapidly from its tangent than the circum- 
ference which is described with the greater radius ; and 
hence we say, that its curvature is greater. And gener- 
ally, the curvature of any curve is said to be greater or less 
than that of another curve, according as its tendency to 
depart from its tangent at a given point, is greater or less 
than that of the curve with which it is compared. 

160. The curvature is the same at all the points of the 
same circumference, and also in all circumferences described 
with equal radii, since the tendency to depart from the tan- 
gent is the same. In different circumferences, the curva- 
ture is measured by the angle formed by two radii drawn 
through the extremities of an arc of a given length. 

Let r and r 1 designate the radii of two circles, a the 
length of a given arc measured on the circumference of 
each ; c the angle formed by the two radii drawn through 
the extremities of the arc in the first circle, and d the 
angle formed by the corresponding radii of the second. 
We shall then have 



2*r : a :: 360° : c, hence, c = 25^ ; 



also, 



2*/ : a :: 360° : </, hence, c' = ^lf 

2*r / 



and consequently 



r r 



156 



ELEMENTS OF THE 



that is, the curvature in different circumferences varies 
inversely as the radii. 

161. The curvature 
of plane curves is meas- 
ured by means of the 
oscillatory circle. 

If we assume two 
points P and P f , either 
on the same or on dif- 
ferent curves, and find 

the radii r and / of the circles which are osculatory at 
these points, then 

1 l_ 

r '' r' ; 




curvature at P : curvature at P' 



that is, the curvature at different points varies inversely 
as the radius of the osculatory circle. 

The radius of the osculatory circle is called the radius 
of curvature. 

162. Let us now determine the value of the radius of 
curvature for lines of the second order. 

The general equation of these lines (An. Geom. Bk. VI % 
Prop. XII, Sch. 3), is 



y 2 = mx + nx 2 , 



which gives, 



d v = ( m + 2nx ) doc do( 2 , d 2 = [4y 2j t-(ni + 2noLf]dx 2 
J 2y ' V 4z/ 2 

72 _ ^ n y d% 2 — ( m + 2 nx) dx dy _ [4 ny 2 — [m + 2 ?ix) 2 ] dx 2 



DIFFERENTIAL CALCULUS. 157 

Substituting these values in the equation 



(da*+dy 2 Y 



R=- 

cixay 

we obtain 

p _ [4:(mx 4- nx 2 ) -\-(m-\- 2nx) 2 ]' i 
J.X ^^ ■- 



which is the general value of the radius of curvature in 
lines of the second order, for any abscissa x. 

163. If we make x = 0, we have 

that is, in lines of the second order, the radius of curva- 
ture at the vertex of the transverse axis is equal to half 
the parameter of that axis. 

■ If be required to find the value of the radius of curva- 
ture at the extremity of the conjugate axis of an ellipse, 
we make (An. Geom. Bk. VIII, Prop. XXI, Sch. 3), 

2J5 2 B 2 , 

m = — — , n = -5 , and x = A, 

A A 2 

which gives, after reducing, 

hence, the radius of curvature at the vertex of the conju- 
gate axis of an ellipse is equal to half the parameter of 
that axis. 

In the case of the parabola, in which n — 0, the general 
value of the radius of curvature becomes 



158 



ELEMENTS OF THE 



R = 



(m 2 + 4 ma?) 2 

2 m 2 



164. If we compare the value of the radius of curvature 

with that of the normal line found ir (Art. 118), we shad 

have 

(normal) 3 



R 



1 



mr 



that is, the radius of curvature at any point is equal to 
the cube of the normal divided by half the parameter 
squared : and hence, the radii of curvature at different 
points of the same curve are to each other as the cubes of 
the corresponding normals. 

Of the Evolutes of Curves. 



165. If we suppose an os- 
culatory circle to be drawn at 
each of the points of the 
curve APP'B, and then a 
curve AGO ' C" to be drawn 
through the centres of these 
circles, this latter curve is 
called the evolute curve, and 
the curve APP'B the invo- 
lute. 



166. The co-ordinates of the centre of the osculatory 
circle, which have been represented by « and /3, are con- 
stant for given values of the co-ordinates x and y of the 




DIFFERENTIAL CALCULUS. 159 

involute curve, but they become variable when we pass 
from one point of the involute curve to another. 

167. We have already seen that the oscillatory circle is 
characterized by the equations (Art. 154) 

(x-«Y + (y-?Y=:R\ (1) 
(x-*)dx±(y-P)dy = 0, (2) 
dx 2 + dy 2 + (y-P)d 2 y = 0. (3) 

If it be required to find the relations between the co- 
ordinates of the involute and the co-ordinates of the 
evolute curves, we must differentiate equations (1) and (2) 
under the supposition that * and £, as well as x and y, 
are variables. We shall then have 

{x — *)dx + (y — P) dy -(x — u)dcc-{y- p) dp = RdR, 

dx 2 + dy 2 + {y — P)d 2 y — docdx — dpdy = 0. 

Combining these with equations (2) and (3), we obtain 

- {y - p)dp - {x - *)d* = RdR } (4) 

— d*>dx — dp dy = 0. 

The last equation gives 

dp _ dx r 

T*~~~ay (5) 

But equation (2) may be placed under the form 

which represents a normal to the involute (Art. 1 22), and 

which becomes, by substituting for — — its value —, 

n d V d « 



160 ELEMENTS OF THE 

y_ j 8 = g(*--) ) (6) 

or (j_« = ^(«_a,) (Art. 120). 

This last equation, which is but another form for the 
equation of the normal to the involute, is, in fact, the 
equation of a tangent line to the evolute, at the point 
whose co-ordinates are a and/3; hence, a normal line to 
the involute curve is tangent to the evolute. 

168. It is now proposed to show, that the radius of cur- 
vature and the evolute curve have equal differentials. 
Combining equations (2) and (5) we obtain 

(*_.) = (y_*)g, (7) 

or by squaring both members, 



.2 



(«_.y = (j,_tf- ; 



combining this last with equation (1) we have 
<^+*!) (y - ,)> = *>. (8) 
Combining equations (4) and (7), we have 
- { y-H)dfi-(y-fi)*L = RdR, 

_(Jf + d£l {y _, ) = RdR: 



DIFFERENTIAL CALCULUS. 161 

or by squaring both members 

Dividing this last by equation (8), member by member, 

we have 

(dltf = d* 2 + dfi 2 

or dR = Vd* 2 + d/3 2 . 

But if s represents the arc of the evolute curve, of which 
the co-ordinates are * and /s, we shall have (Art. 128), 



ds = Vd* z + c//2 2 ; 

hence, dR = ds; 

that is, 2Ae differential of the radius of curvature is equal 
to the differential of the arc of the evolute. 

169. It does not follow, however, from the last equation, 
that the radius of curvature is equal to the arc of the evolute 
curve, but only that one of them is equal to the other plus 
or minus a constant (Art. 22). Hence, 
R = s + a 

is the form of the equation which expresses the relation 
between them. 



162 



ELEMENTS OF THE 



If we determine the radii 
of curvature at two points of 
the involute, as P and P ; , 
we shall have, for the first, 

R = s + a, 

and for the second 



R f = s' + a 



hence, 



R'-R = s f 



C'C"; 




and hence, the difference between the radii of curvature at 
any two points of the involute is equal to the part of the 
evolute curve intercepted between them. 

170. The value of the constant a will depend on the 
position of the point from which the arc of the evolute 
curve is estimated. 

If, for example, we take the radius of curvature for lines 
of the second order, and estimate the arc of the evolute 
curve from the point at which it meets the axis, the value 
of s will be when R = — m (Art. 163): hence we 
shall have 



— m = + a or a 

fit 



■m\ 



and for any other point of the curve 






DIFFERENTIAL CALCULUS. 



163 



Either of the evolutes, FE, 
FE', FE', or F'E, corres- 
ponding to one quarter of the 
ellipse, is equal to (Art. 169) 



B 



A 




171 . The evolute curve takes 
its name from the connexion which it has with the corres- 
ponding involute. 

Let CC'C" be an evolute 
curve. At C draw a tan- 
gent AC, and make it equal 
to the constant a in the equa- 
tion 

R = s + a. 

Wrap a thread ACQ ' C" 
around the curve, and fasten 
it at any point, as C" . 

Then, if we begin at A, 
and unwrap or evolve the 
thread, it will take the positions PC, P'C", &c, and the 
point A will describe the involute APP f : for 

PC'-AC= CO and P'C" - AC = CC'C", &c 

172. The equation of the evolute may be readily found 
by combining the equations 







dy(da?+dy 2 ) 
dxcPy 



with the equation of the involute curve. 



164 ELEMENTS OF THE 

1st. Find, from the equation of the involute, the values of 
— and d?y, 

CLX 

and substitute them in the two last equations, and there 
will be obtained two new equations involving «, /3, x and y. 
2d. Combine these equations with the equation of the 
involute, and eliminate x and y : the resulting equation 
will contain *, /3, and constants, and will be the equation 
of the evolute curve. 

173. Let us take, as an example, the common parabola 
of which the equation is 



y 2 = mx. 



We shall then have 



dy _ m „ __ m 2 da? 

Tx'Ty' y ~~"Ty r ' 

and hence 

a - & - 4 y 3 / 4y 2 + m 2 \ _ 4?/ 3 + m 2 y _ 4y # 

V P ~ m 2 \ \y 2 )~ m 2 ~m 2 ^ V> 

and by observing that the value of x — * is equal to that 
of y — j3 multiplied by — f- y we have 



4 if + #r 

2m ' 



hence we have, 



4y 3 . 2y 2 m 

£ = -^r- and # — * = - : 

mr m 2 



D1F ERENTIAL CALCULUS- 165 

substituting for y its value in the equation of the involute 



we obtain 



y — m^x*, 



3^ 

4# 2 _ m 

p = — r , x-« = — 2x- — ; 



and by eliminating x, we have 

.(a mY 

\ 2 / 



/3 2 = 

27 m^ 



which is the equation of the evolute. 
If we make /3 = 0, we have 

1 

2 ' 

and hence, the evolute meets the 
axis of abscissas at a distance from 
the origin equal to half the param- 
eter. If the origin of co-ordinates 
be transferred from A to this 
point, we shall have 




and consequently 



2 ' 



27 m 



The equation of the curve shows that it is symmetrical 
with respect to the axis of abscissas, and that it does not 
extend in the direction of the negative values of <*>'. The 
evolute CO corresponds to the part AP of the involute, 
and CC" to the part AP' . 



166 ELEMENTS OF THE 



CHAPTER VIII. 

Of Transcendental Curves. — Of Tangent Planes 
and Normal Lilies to Surfaces. 

174. Curves may be divided into two general classes : 
1st. Those whose equations are purely algebraic ; and 
2dly. Those whose equations involve transcendental 

quantities. 

The first class are called algebraic curves, and the 
second, transcendental curves. 

The properties of the first class having been already 
examined, it only remains to discuss the properties of the 
transcendental curves. 

Of the Logarithmic Curve. 

175. The logarithmic curve takes its name from the 
property that, when referred to rectangular axes, one of 
the co-ordinates is equal to the logarithm of the other. 

If we suppose the logarithms to be estimated in paral- 
lels to the axis of Y, and the corresponding numbers to 
be laid off on the axis of abscissas, the equation of the 
curve will be 

y = lx. 



DIFFERENTIAL CALCULUS. 



167 




176. If we designate the 
base of a system of loga- 
rithms by a, we shall have, 
(Alg. Art. 241) 



and if we change the value 
of the base a to a', we shall 
have 



It is plain, that the same value of x, in the two equations, 
will give different values of y, and hence, every system of 
logarithms will give a different logarithmic curve. 

If we make y — 0, we shall have (A.]g. Art. 257) 
x — 1 ; and this relation being independent of the base of 
the system of logarithms, it follows, that every logarithmic 
curve will intersect the axis of numbers at a distance from 
the origin equal to unity. 

The equation 

a y = x, 

will enable us to describe the curve by points, even with- 
out the aid of a table of logarithms. For, if we make 



y = 0, y = 



1 



y 



&c. 



we shall find, for the corresponding values of x, 

a? = l, x—i/a, x = a-y/a, x = tfa &c. 

177. If we suppose the base of the system of logarithms 
to be greater than unity, the logarithms of all numbers less 



168 ELEMENTS OF THE 

than unity will be negative (Alg. Art. 256) ; and therefore, 
the values of y corresponding to the abscissas, between the 
limits x = and x — AE = 1, will be negative. Hence, 
these ordinates are laid off below the axis of abscissas. 

When x = 0, y will be infinite and negative (Alg. Art. 
264). If we make x negative, the conditions of the equa- 
tion cannot be fulfilled ; and hence, the curve does not 
extend on the side of the negative abscissas. 

178. Let us resume the equation of the curve 

y = Ix. 

If we represent the modulus of the system of logarithms 
by A, and differentiate, we obtain (Art. 56), 

dx 



dy = A 



x 



dy _ A 
dx x 

But -j- represents the tangent of the angle which the 

tangent line forms with the axis of abscissas : hence, the 
tangent will be parallel to the axis of abscissas when 
x — qo , and perpendicular to it when x = 0. 

But when x = 0, y = — oo ; hence, the axis of ordinates 
is an asymptote to the curve. The tangent which is 
parallel to the axis of X is not an asymptote : for when 
x — oo , we also have y = oo . 

179. The most remarkable property of this curve be- 
longs to its sub-tangent VR', estimated on the axis of 
logarithms. We have found, for the sub-tangent, on the 
axis of X (Art. 114), 



DIFFERENTIAL CALCULUS. 169 

dx 
and by simply changing the axes, we have 

ax 

hence, the sub-tangent is equal to the modulus of the 
system of logarithms from which the curve is constructed. 
In the Naperian system M = l 3 and hence the sub-tangent 
will be equal to 1 = AE. 



Of the Cycloid. 




180. If a circle NPG be rolled along a straight line 
AL, any point of the circumference will describe a curve, 
which is called a cycloid. The circle NPG is called the 
generating circle, and P the generating point. 

It is plain, that in each revolution of the generating circle 
an equal curve will be described ; and hence, it will only 
be necessary to examine the properties of the curve 
APBL, described in one revolution of the generating circle. 
We shall therefore refer only to this part when speaking 
of the cycloid. 

181. If we suppose the point P to be on the line AL 
at A y it will be found at some point, as L, after all. the 



170 



ELEMENTS OF THE 



G 1 


^—~ ^/ 


\ pLl...... 


° ) 


^ 


\ / 


yk 


V 


V 


V 



A R N 



M 



points of the circumference shall have been brought in 
contact with the line AL. The line AL will be equal to 
the circumference of the generating circle, and is called 
the base of the cycloid. The line BM, drawn perpen- 
dicular to the base at the middle point, is equal to the 
diameter of the generating circle, and is called the axis of 
the cycloid, 

182. To find the equation of the cycloid, let us assume 
the point A as the origin of co-ordinates, and let us sup- 
pose that the generating point has described the arc AP. 
If N designates the point at which the generating circle 
touches the base, AN will be equal to the arc NP. 

Through N draw the diameter NG, which will be 
perpendicular to the base. Through P draw PR perpen- 
dicular to the base, and PQ parallel to it. Then, PR = NQ 
will be the versed-sine, and PQ the sine of the arc NP. 

Let us make 

ON = r, AR = x, PR = NQ = y, 

we shall then have 



PQ= i/2ry — tf 9 x = AN - RN = aicNP - PQ: 

hence, the transcendental equation is 



x = ver- sin l y — y 2 ry — y 



DIFFERENTIAL CALCULUS. 171 

183. The properties of the cycloid arc, however, most 
easily deduced from its differential equation, which is 
readily found by differentiating both members of the trans- 
scendental equation. 

We have (Art. 71), 

d(xev-sin~ l y) = — , 

V2nj-f 



V2 



hence, 



ry-y* 



dx 



rdy rdy — ydy 



V2 ry — y 2 -y/2 ry — y 2 



or dx 



ydy 



V2ru — 



y - y 

which is the differential equation of the cycloid. 

184. If we substitute in the general equations of (Arts. 
114, 115, 116, 117), the values of dx, dy, deduced from 
the differential equation of the cycloid, we shall obtain the 
values of the normal, sub-normal, tangent, and sub-tangent. 
They are, 

normal PN — -\/2ry, sub-normal RN — V2ry — y 2 , 



tangent 



p T = J L}^riZ , sub-tangent TR=~- 



V2ry — y 2 -\/2ry — y 2 

These values are easily constructed, in consequence of 
their connexion with the parts of the generating circle. 

The sub-normal RN, for example, is equal to PQ of 
the generating circle, since each is equal to ^/2ry — y 2 : 
hence, the normal PN and the diameter GN intersect 
the base of the cycloid at the same point. 



172 ELEMENTS OF THE 

Now, since the tangent to the cycloid at the point P is 
perpendicular to the normal, it must coincide with the 
chord PG of the generating circle. 

If, therefore, it be required to draw a normal or a tan- 
gent to the cycloid, at any point as P, draw any line, as 
ng, perpendicular to the base AL, and make it equal to 
the diameter of the generating circle. On ng describe a 
semi-circumference, and through P draw a parallel to the 
base of the cycloid. Through p, where the parallel cuts 
the semi-circumference, draw the supplementary chords 
pn, pg, and then draw through P the parallels PN, PG, 
and PN will be a normal, and PG a tangent to the cycloid 
at the point P. 

185. Let us resume the differential equation of the 
cycloid 



dx = 



ydy 



V2ry — y 2 
which may be put under the form 



dy = V2n/-z/ 2 /2F 
dx y V y 

If we make y = 0, we shall have 

dy 

and if we make y = 2r, we shall have 

f = 0: 

dx 



DIFFERENTIAL CALCULUS. 3 73 

hence, the tangent lines drawn to the cycloid at the points 
where the curve meets the base, are perpendicular to the 
base; and the tangent drawn through the extremity of the 
greatest ordinate, is parallel to the base. 
186. If we differentiate the equation 



dx 



y d y 



-\/2ry — y 2 
regarding dx as constant, we obtain 

V2ry — y 2 
or by reducing and dividing by y, 

= (2ry - y 2 )d?y + rdy 2 , 

whence we obtain 

J 2ry — y 2 

and hence the cycloid is concave towards the axis of 
abscissas (Art. 133). 

187. To find the evolute of the cycloid, let us first sub- 
stitute in the general value of 

R _ (da> + dy*p 

dxd?y 

the value of d?y found in the last article : we shall then 
have 



R = 2 i {ryY = 2V2ry: 

hence, the radius of curvature corresponding to the ex- 
tremity of any ordinate y, is equal to double the normal. 



174 



ELEMENTS OF THE 



The radius of curvature is when y = 0, and equal to 
twice the diameter of the generating circle for y = 2r : 
hence, the length of the evolute curve from A to A' is 
equal to twice the diameter of the generating circle. 

Substituting the value of cPy in the values of y— /3, 
x — cc (Art. 172), we obtain 



y — t*=z2y, x — tt— — 2 V'2ry — y 2 ; 
hence we have 



-A 



^ = *-2V-2r/3-^ 2 . 



Substituting these values of y and x in the transcen- 
dental equation of the cycloid, we have 



cc — ver-sm 



|8+ V-2r/3- 



which is the transcendental equation of the evolute, re- 
ferred to the primitive origin and the primitive axes. 

Let us now trans- 
fer the origin of co- 
ordinates to the point 
A f , and change at 
the same time the 
direction of the posi- 
tive abscissas : that 
is, instead of estima- 



ting them from the X' 
left to the right, we will estimate them from the right 
to the left. Let us designate the co-ordinates of the 
evolute, referred to the new axes A! M, A'X', bv *' and ,fl ; 




DIFFERENTIAL CALCULUS. 175 

Since A! X ' = AM = the semi-circumference of the gene- 
rating circle, which is equal to rw, we shall have, for the 
abscissa A'R' of any point P ; , 

A f R f = * f — r* — *, hence, * = rw — *f : 

and for the ordinate, we shall have 

R!P> ' = ?'= RE - P E == 2r- (- j3) = 2r + /3, 

hence, p = — 2r + £', or — /3 = 2?" — & . 

Substituting these values of * and /3 in the transcen- 
dental equation of the evolute, we obtain 



T7? — ct z= ver-sin 



-i(2r-/30+V2r^-^ 2 , 



or 



*f=7ir— ver-sin" 1 (2 r — /3') — V2r/3'— /3 /2 



But the arc whose versed-sine is 2r — .fl', is the supple- 
ment of the arc whose versed-sine is p', hence 



a! — ver-sin ! j3' — V2r(Z f — /3 /2 , 

which is the equation of the evolute referred to the new 
origin and new axes. 

But this equation is of the same form, and involves the 
same constants as that of the involute : hence, the evolute 
and involute are equal curves. 



Of Spirals. 

188. A spiral is a curve described by a point which 

moves along a right line, according to any law whatever, 

the line having at the same time a uniform angular motion. 

12 



176 



ELEMENTS OF THE 




Let ABC be a straight 
line which is to be turned 
uniformly around the 
point A. When the 
motion of the line be- 
gins, let us suppose a 
point to move from A 
along the line in the 
direction ABC. When 
the line takes the posi- 
tion ADE the point will 

have moved along it to some point as D, and will have 
described the arc AaD of the spiral. When the line 
takes the position AD'E 1 the point will have described 
the curve AaDD', and when the line shall have comple- 
ted an entire revolution the point will have described the 
curve AaDUB. 

The point A, about wdiich the right line moves, is 
called the pole ; the distances AD, AD', AB, are called 
radius-vectors, and if the revolutions of the radius-vector 
are continued, the generating point will describe an in- 
definite spiral. The parts AaDD'B, BFF'C, described in 
each revolution, are called spires. 

189. If with the pole as a centre, and AB, the distance 
passed over by the generating point in the direction of the 
radius-vector during the first revolution, as a radius, we 
describe the circumference BEE', the angular motion of 
the radius-vector about the pole A, may be measured by 
the arcs of this circle, estimated from B. 

If we designate the radius-vector by u, and the measur- 
ing arc, estimated from B, by t, the relation between u 



DIFFERENTIAL CALCULUS. 177 

and t, may in general be expressed by the equation 

u — at n , 

in which n depends on the law according to which the 
generating point moves along the radius-vector, and a on 
the relation which exists between a given value of u and 
the corresponding value of t. 

190. When n is positive the spirals represented by the 
equation 

u = aV, 

will pass through the pole A. For, if we make £ = 0, we 
shall have u — 0. 

But if n is negative, the equation will become 

a 

u = at n , or u = — , 

in which we shall have 

u — oo for t = 0, 

and u = for t = oo : 

hence, in this class of spirals, the first position of the 
generating point is at an infinite distance from the pole : 
the point will then approach the pole as the radius-vector 
revolves, and will only reach it after an infinite number of 
revolutions. 

191. If we make n = 1, the equation of the spiral be- 
comes 

u = at. 

If we designate two different radius -vectors by v! and 
u ff , and the corresponding arcs by tf and tf f , we shall have 

v! = at\ and u" == at f/ , 



178 



ELEMENTS OF THE 



and consequently 



that is, the radius-vectors are proportional to the measur- 
ing arcs, estimated from the point B. This spiral is 
called, the spiral of Archimedes. 

192. If we represent by unity the distance which the 
generating point moves along the radius-vector, during one 
revolution, the equation 

u = at, 
will become 



1 -at, 



or 



1 X — = t. 
a 



But since t is the circumference of a circle whose 
radius is unity, we shall have 

— = 2t, and consequently, a = ■ 
a 

193. If the axis BD, of 
a semi-parabola BCD, be 
wrapped around the circum- 
ference of a circle of a 
given radius r, any abscissa, 
as Bb, will coincide with 
an equal arc Bb', and any 
ordinate as ba, will take the 
direction of the normal Ab'a' . 
The curve Ba'c', described 

through the extremities of the ordinates of the parabola, is 
called the parabolic spiral. 

The equation of this spiral is readily found, by observing 
that the squares of the lines b'a! , c d , &c, are propor- 
tional to the abscissas or arcs Bb', Be . 




DIFFERENTIAL CALCULUS. 179 

If we designate the distances, estimated from the pole 
A, by u, we shall have l/a' = u — r : hence, 

(u — rf = 2pt, 

is the equation of the parabolic spiral. 

If we suppose r = 0, the equation becomes 

u 2 — 2pt. 

If we make n= — 1, the general equation of spirals 
becomes 

u = at~ l , or ut = a. 

This spiral is called the hyperbolic spiral, because of the 
analogy which its equation bears to that of the hyperbola, 
when referred to its asymptotes. 

194. The relation between u and t is entirely arbitrary, 
and besides the relations expressed by the equation 

u = af, 
we may, if we please, make 

t = \ogu. 

The spiral described by the extremity of the radius-vec- 
tor when this relation subsists, is called the logarithmic 
spiral. 

195. If in the equation of the hyperbolic spiral, we 
make successively, 

,i l l l * 

t=h =2> =3' = 4' &C -' 

we shall have the corresponding values, 

u = a, u = 2a, u = Sa, w = 4a, &c. 



180 



ELEMENTS OF THE 



Through the 
pole A draw AD 
perpendicular to 
AB, and make 
it equal to a : 
then through D 
draw a parallel 
to AB. From 
any point of the 

spiral as P draw PM perpendicular to AB, we shall 
then have 

PM = u sin MAP = u sin t. 




If we substitute for u its value — , we shall have 

t 

PM=a . 

t 

Now as the arc t diminishes, the ratio of will ap- 
proach to unity, and the value of the ordinate PM will 
approach to a or CM: hence, the line DC approaches 
the curve and becomes tangent to it when t = 0. But 
when t = 0, u = oo ; hence, the line DC is an asymptote 
of the curve. 

196. The arc which measures the angular motion of the 
radius-vector has been estimated from the right to the left, 
and the value of t regarded as positive. If we i evolve 
the radius-vector in a contrary direction, the measuring 
arc will be estimated from left to right, the sign of t will 
be changed to negative and a, similar spiral will be de- 
scribed. The line DC is an asymptote to the hyperbolic 
spiral, corresponding to the negative value of t. 



DIFFERENTIAL CALCULUS- 



181 



197. Let. us now find a general value for the subtan- 
gent of any curve referred to polar co-ordinates. The 
subtangent is the projection of the tangent on a line 
drawn through the pole and perpendicular to the radius- 
vector passing through the point of contact. 

The equation of the curve may be written under the 

form 

u =f(t), 

in which we may suppose t the independent variable, and 
its first differential constant. 

Let AO =1 be the radius of 
the measuring circle, PT a tan- 
gent to the curve at the point P, 
and A T drawn perpendicular to 
the radius-vector AP, the sub- / 
tangent. 

Take any other point of the 
curve as P', and draw AP' . 
About the centre A describe the 
arc PQ, and draw the chord PQ. 
Draw also the secant PP' and 
prolong it until it meets AT ; , 
drawn parallel to QP, at T'. 

From the similar triangles QPP', A T'P', we have 

PQ : QP' :: AT' : AP' ; 
QP' AP' 




hence 



PQ AT'' 



But when we pass to the limit, by supposing the point 
P' to coincide with P, the secant VPP' will become the 
tangent PT, and AT will become the subtangent AT. 



182 



ELEMENTS OF THE 



, ..£ 



But under this supposition 
the arc NN' will become equal 
to dt, the arc PQ to the chord 
PQ (Art. 128), AP' to u, and 
the line QP f to du. 

To find the value of the arc 
PQ, we have 

1 : NN' : : AP : arc PQ; 

hence, 

1 : dt : : u : arc PQ, 

and PQ = w<fr. 

Substituting these values, and passing to the limit, we 
have 

du u 

~uTt~1T : 

hence, we have the subtangent 




AT = 



uHt 
du 



198. If we find the value of u 2 and du from the gen- 
eral equation of the spirals 



u = at*, 



we shall have 



AT=-t" + \ 
n 



DIFFERENTIAL CALCULUS. 183 

In the spiral of Archimedes, we have 
n = 1 , and a = — ; 

t 2 
hence, AT — — . 

2* 

If now we make t = 2^ = circumference of the mea- 
suring circle, we shall have 

AT= 2n = circumference of measuring circle* 

After m revolutions, we shall have 

t = 2 mTr, 
and consequently, 

AT= 2m 2 7r — m.2m7r; 

that is, the subtangent, after m revolutions, is equal to 
m times the circumference of the circle described with 
the radius-vector. This property was discovered by 
Archimedes. 

199. In the hyperbolic spiral n= — 1, and the value of 
the subtangent becomes 

AT= -a; 

that is, the subtangent is constant in the hyperbolic spiral. 

200. It may be remarked, that 

AT _udt 
AP ~ du 

expresses the tangent of the angle which the tangent makes 
with the radius-vector. 



184 ELEMENTS OF THE 

In the logarithmic spiral, of which the equation is 
t=\ogu, 

we have dt = A — ; 

u 

. A T udt 

lience - AP=-d^ =A; 

that is, in the logarithmic spiral, the angle formed by the 
tangent and the radius-vector passing through the point of 
contact, is constant; and the tangent of the angle is equal 
to the modulus of the system of logarithms. If t is the 
Naperian logarithm of u, the angle will be equal to 45°. 

201. The value of the tangent in a curve referred to 
polar co-ordinates, 



PT=\/AP 2 + A'f = u\/l + 



u 2 dt 



du z ' 

202. To find the differential of the arc, which we will 
represent by z, we have 

PP' = ^QF 2 +QP 2 ; 

or, by substituting for QP f and PQ their values, and 
passing to the limit, we have 

dz=Vdu 2 + u 2 dt*. 



DIFFERENTIAL CALCULUS. 



185 



203. The differential of the 
area ADP when referred to the 
polar co-ordinates, is not an ele- 
mentary rectangle as when re- 
ferred to rectangular axes, but ; ; 
is the elementary sector APP'. \ 

The limit of the ratio of the 
sector APP' with the arc NN f , 
will be the same as that of 
either of the sectors APQ, 
AP"P' between which it is 
contained, with the same arc 
NN f . Hence, if we designate 
the area by s, and pass to the limit, we shall have 




ds 
dt 



APxPQ 
2NN' 



or 



ds 



u 2 dt 



which is the differential of the area of any segment of a 
spiral. 



Of Tangent Planes and Normal Lines to Surfaces. 

204. Let u = F(cc,y,z) = Q, 

be the equation of a surface. 

If through any point of the surface two planes be passed 
intersecting the surface in two curves, and two straight 
lines be drawn respectively tangent to each of the curves, 
at their common point, the plane of these tangents will be 
tangent to the surface. 

205. Let us designate the co-ordinates of the point at 
which the plane is to be tangent by a/' } y" , z" . 



186 ELEMENTS OF THE 

Through this point let a plane be passed parallel to the 
co-oidinate plane YZ. This plane will intersect the 
surface in a curve. The equations of a straight line tan- 
gent to this curve, at the point whose co-ordinates are 
rf! , ,y",z", are 

* = */>= a", y-y" = ^Z-z"); 

the first equation represents the projection of the tangent 
on the co-ordinate plane ZX, and the second its projec- 
tion on the co-ordinate plane YZ (An. Geom. Bk. IX. 
Art. 70). 

Through the same point let a plane be passed parallel to 
the co-ordinate plane ZX, and we shall have for the 
equations of a tangent to the curve 

y = y»=V\ x-a/'^Xz-z"); 

The coefficient -j- represents the tangent of the angle 

which the projection of the first tangent on the co-ordinate 

plane YZ makes with the axis of Z ; and the coefficient 

dx 

— represents the tangent of the angle which the projection 

dz 

of the second tangent on the plane ZX makes with the 
axis of Z (An. Geom. Bk. VIII, Prop. II). 

But these coefficients may be expressed in functions of 
the surface and the co-ordinates of its points. For, we 

have 

u = f(x,y,z) = 0, 

and if we suppose x constant, we shall have (Art. 87) 

du = -r dy + — dz = : 
dy J dz 



DIFFERENTIAL CALCULUS. 187 

du 

, du dz 

hence, -^ = — : 

dz du 

dy 

and if we suppose y constant, we shall find, in a similar 
manner, 

du 

dx _ dz 

dz ~ du 

dx 

hence, the equation of the projection of the first tangent on 
the plane of YZ becomes 

du 





y-y"= 


dz , 
dy 


-*"); 






and the equation of the 


projection 


of the 


second 


tangent 


on the plane 


of ZX is 


du 










x — x" — 


dz , 
dx 


- *")• 







The equation of a plane passing through the point whose 
co-ordinates are x", y f/ , z" is of the form 

A (x -a/0 + B{y - y") + C(s - z") = 0, 

C 
in which — —will represent the tangent of the angle which 
B 

the trace on the co-ordinate plane YZ makes with the 

Q 

axis of Z, and — 7~ tne tangent of the angle which the 

A. 

trace on the plane of ZX makes with the axis of Z. 



188 ELEMENTS OF THE 

But since the tangents are respectively parallel to the 
co-ordinate planes YZ, ZX, their projections will be 
parallel to the traces of the tangent plane : therefore, 

du 





du 


c 


Tz 





__ i 


B 


du 




dy 




du 


C 


dz 


A ~ 


du' 




dx 



hence, - 


- B = 


du 

dz 

du 
dx c 

du 


hence, - 


- A = 



dz 

Substituting these values of B and A in the equation 
of the plane, and reducing, we find 

which is the equation of a tangent plane to a surface at a 
point of which the co-ordinates are x", y f/ , z" . 

206. A normal line to the surface being perpendicular 
to the tangent plane at the point of contact, its equations 
will be of the form 

du du 

dz dz 



ELEMENTS 



OF THE 



INTEGRAL CALCULUS. 



Integration of Differential Monomials. 

207. The Differential Calculus explains the method of 
finding the differential of a given function. The Integral 
Calculus is the reverse of this. It explains the method 
of finding the function which corresponds to a given 
differential. 

The rules for the differentiation of functions are explicit 
and direct. Those for determining the integral, or func- 
tion, from the differential expression, are less direct and 
are deduced by reversing the process by which we pass 
from the function to the differential. 

208. Let it be required, as a first example, to integrate 
the expression. 

x m dx. 

We have found (Art. 32), that 

d{x m + l )={m + l)x m dx, 

dx m+l ,/r\ 

whence, xax — — = a [ ) , 

m + 1 Vm + 1/ 



190 ELEMENTS OF THE 

.r m+1 
and consequently , 

is the function of which the differential is x m dx. 

The integration is indicated by placing the character / 
before the differential which is to be integrated. Thus, 
we write 

f x m dx=~ . 

J m + l' 

from which we deduce the following rule. 

To integrate a monomial of the form x m dx, augment 
the exponent of the variable by unity, and divide by the 
exponent so increased and by the differential of the 
variable. 

209. The characteristic / signifies integral or sum. 
The word sum, was employed by those who first used the 
differential and integral calculus, and who regarded the 
integral of 

x m dx 

as the sum of all the products which arise by multiplying 
the mth power of x, for all values of x, by the con- 
stant dx. 

dx 

210. Let it be required to integrate the expression — . 

or 



We have, from the last rule, 




P (XJC /* 7 —-3 *•*•' *** 


1 


. ^ x 3 ~ ^ ~_3-r-l~-2~ 


2x~ 2 


In a similar manner, we find 




1 5 

fdx y/x 2 — fx J dx = = = 

- + 1 — 
3 3 


6 
3#3 

5 ' 



INTEGRAL CALCULUS. 191 

211. It lias been shown (Art. 22), that the differential 
of the product of a variable multiplied by a constant, is 
equal to the constant multiplied by the differential of the 
variable. Hence, we may conclude that, the integral of 
the product of a differential by a constant, is equal to the 
constant multiplied by the integral of the differential : 
that is, 



J ax m dx = afx m dx 



x 
a 



m + \ 



Hence, if the expression to be integrated have one or 
more constant factors, they may be placed as factors with- 
out the sign of the integral. 

212. It has also been shown (Art. 22), that every con- 
stant quantity connected with the variable by the sign 
olus or minus, will disappear in the differentiation ; and 
hence, the differential of a -f x m , is the same as that of 
x m ; viz. mx m ~ l dx. Consequently, the same differential 
may answer to several integral functions differing from 
each other in the value of the constant term. 

In passing, therefore, from the differential to the integral 
or function, we must annex to the first integral obtained, 
a constant term, and then find such a value for this term 
as will characterize the particular integral sought. 

For example (Art. 94), 

-JL = a, or dy = adx, 
ax 

is the differential equation of every straight line which 

makes with the axis of abscissas an angle whose tangent 

is a. Integrating this expression, we have 

13 



192 ELEMENTS OF THE 

fdy — af dec, 
or y — ax, 

or finally, y = ax + C. 

If now, the required line is to pass through the origin 
of co-ordinates, we shall have, for 

x = 0, y = 0, and consequently, C — 0. 

But if it be required that the line shall intersect the axis 
of Y at a distance from the origin equal to + b, we shall 
have, for 

x = 0, y — +b, and consequently, C .== + 6 ; 

and the true integral will be 

y = ar + b. 

If, on the contrary, it were required that the right line 
should intersect the axis of ordinates below the origin, we 
should have, for 

x — 0, y = — b, and consequently, C = — 6 ; 
and the true integral would be 

y = az — b. 
213. It has been shown (Art. 95), that 
xdx + ydy = 

is the differential equation of the circumference of a circle. 
By taking the integral, we have 

/ xdx -f / ydy = 0, or x 2 +2Z 2 = 0? 

or finally, ^ -f ^ + C = 0. 



INTEGRAL CALCULUS. 193 

If it be required that this integral shall represent a given 
circumference, of which the radius is R, we shall have, 
by making 

x = 0, y 2 = - C = R 2 , 
and hence, C = — R 2 ; 

and consequently the true integral is 

a? + y 2 -R 2 = 0, or x 2 + y 2 r= R\ 

The constant C, which is annexed to the first integral 
that is obtained, is called an arbitrary constant, because 
such a value is to be attributed to it as will cause the 
required integral to fulfil given conditions, which may be 
imposed on it at pleasure. 

The value of the constant must be such, as to render 
the equation true for every value which can be attributed 
to the variables. 

214. There is one case to which the formula of Art. 208 
does not apply. It is that in which m = — 1. Under this 
supposition, 



f T *J _ X _ X _ X _ 


1 

-=00. 


/ U, UAL — -' - - "- — — _ 

J ra-fl -1 + 1 


when m = — 1, 




foc m dx = fx- x dx= / — , 





and 



f—=:\ogx+C. (Art. 57). 
*j x 



215. Since the differential of a function composed of 
several terms, is equal to the sum or difference of the diffe- 
rentials (Art. 27), it follows that the integral of a differen- 



194 ELEMENTS OF THE 

tial expression, composed of several terms, is equal to the 
sum or difference of the integrals taken separately. For 
example, if 

du — adx — — 3- + x y x dx, we have 
x 

fdu —fiadx 3-+ x -y/x~dx\ and 

1 /? 2 £ 

u = ax + -—+—x* + C. 
2ar 5 

216. Every polynomial of the form 

(a + bx + ex 2 -f 6ic.) n dx, 

m which n is a positive and whole number, may be inte- 
grated by the rule for monomials, by first raising the poly- 
nomial to the power indicated by the exponent, and then 
multiplying each term by dx. 

If, for example, we make n = 2, and employ but two 
terms, we have 

f(a 4- bxfdx =f(a 2 dx + 2abxdx + bWdx), 

7,2 3 

= a 2 x + abx 2 + — + C. 
o 

Integration of Particular Binomials. 

217. If we have a binomial of the form 

d,u = (a + bx n ) m x n - l dx\ 

that is, in which the exponent of the variable without the 
parenthesis is less by unity than the exponent of the vari- 
able within, we may make 



INTEGRAL CALCULUS. 195 

a + bx n = z, which gives 

nbx n ~ 1 dx = dz, or x n ~ 1 dx = — ^: 

nb 

d* z m * x 

whence du = z m -^-, or 



nb (m+\)nb' > 

and consequently 

(m+ \)nb 

Hence, the integral of the above form, is equal to the bino-* 
mial factor with its exponent augmented by unity, divided 
by the exponent so increased, into the exponent of the vari- 
able within the parenthesis into the coefficient of the 
variable. 

For example, 

f{a -f 3X 2 ) 3 xdx = (< * + 3 ^ )4 + C; and 

L m 1 

f(a + bx 2 ) 2 mxdx = ^(<z + 6a? 2 ) 2 + C. 

218. A transformation similar to that of the last article 
will enable us to integrate certain differentials correspond- 
ing to logarithmic functions. If we have an expression of 
the form 

adx 



du = 



c + bx* 



dz 
make c-^-bx — z, which gives dx = — t and by sub- 
stituting, we have 

/adx Cadz a f 'dz a , . ~ 



196 ELEMENTS OF THE 

and by substituting for z its value 



A 



a log(c + 6*)+C. 



c + 6a? 6 ° 
In a similar manner, we should find 
adx a 



k 



— bx b 



log (a — bx) + C, 



in which the integral is negative, since d( — x) = — da?. 
. *We can find, in a similar manner, the integral of every 
fraction of which the numerator is equal to the differential 
of the denominator, or equal to that differential multiplied 
by a constant. 

If, for example, we have 

7 (b 4- 2cx)mdx 

du = * ; — 5- ; 

a + bx + car 

make a + bx + ca? 2 = z, which gives, bdx + 2cxdx = dz, 
and hence, 

, mdz 
du = , or u — miogz, 

and by substituting for z its value 

u = m\og(a + bx + ca? 2 ). 

Of Differentials whose Integrals are expressed by 
the Circular Functions. 

219. We have seen, Art. 71, that if a? designates an arc 
and u the sine, to the radius unity, we shall have 

, du 

dx — — T =. . 
Vl-u 2 



INTEGRAL CALCULUS. 197 

du 



hence, / . = x + C ; 

J Vl-u 2 

or adopting the notation of Art. 72, 
du 



/au 



— sin l u+ C. 



If the arc expressed in the second member of the equa- 
tion be estimated from the beginning of the first quadrant, 
the sine will be 0, when the arc is 0, and we shall have, 
for u = 

du 



f- 



= 0, and consequently C = 0, 



Vi 

and under this supposition, the entire integral is 
du . , 



/ 



vr 



To give an example, showing the use of the arbitrary- 
constant, let us suppose that the arc which is to be ex- 
pressed by the second member of the equation, is to be 
estimated from the beginning of the second quadrant. This 
supposition will render 

du 



i 



VT 



U 



for u = 1 , 



But when u = 1, sin" 1 ^ = — n ; hence, 

2 

2- v +C = 0, or C = - — sr : 
2 ' 2 

and we have, for the entire integral, under this supposition, 

du . , 1 



/ 



— ; . . — sin u, — —— 



5T. 



198 



ELEMENTS OF THE 



220. It frequently happens that we have expressions to 
integrate of the form 

dz 



Va 2 -z 2 ' 

Let us suppose, for a moment, that a is the radius of a 
circle, and z the sine of any arc of the circle ; and that u 
is the sine of an arc containing an equal number of degrees 
in a circle whose radius is unity : we shall then have, 



hence, 



: a 



u = — , and 
a 



z : 



du 



dz 



and consequently, 
du 



/du r 

Vu^u~ 2 ~J 



r dz 
a 



L 



v 



/dz 



hence, 



/du r dz 

VT^7 2 ~J Va^z 



sin 



the arc being still taken in a circle whose radius is unity. 

221. We have seen (Art. 71), that if x designates an 
arc, and u the cosine, to the radius unity, we shall have 

du 



dx 



Vl-u 2> 



hence, 



/ 



du 



VT 



= x + C; 



or adopting the notation of Art. 72, 



/- 



du 



VI 



cos x w+ C. 



INTEGRAL CALCULUS. 199 

If the arc be estimated from the beginning of the first 
quadrant, it will be equal to — * for u = 0; hence, the 
first member of the equation becomes equal to — -x when 
u — 0. But under this supposition, cos -1 u~ — n : hence, 
C = 0, and the entire integral is 

r dU _ x 

/ , = COS u. 

Vl-U 2 

222. By a method analogous to that of Art. 220, we 
should find 

r dz yZ 

J 7 = COS — , 

J Va 2 -z 2 a 

the arc being estimated to the radius unity. 

223. We have seen (Art. 71), that if x represents an 
arc, and u its tangent, to the radius unity, we have 

, du 

l + u 2 ' 

hence, / s = x + C : 

J l + u 2 

or, adopting the notation of Art. 72, 

/rT^ =tans " ,M+c - 

If the arc is estimated from the beginning of the first 
quadrant, we shall have 

/* du 
tang" 1 = 0, when l = ; hence, C = 0, 

j 1 -p u 



200 ELEMENTS OF THE 



and the entire integral is 



A 



du 



tang u 



l + u 2 

224. To integrate expressions of the form 
dz 

let us suppose for a moment that a is the radius of a circle, 

and z the tangent of any arc, and that u is the tangent 

of an arc containing an equal number of degrees in a circle 

whose radius is unity : we shall then have, as in 

(Art. 220), 

1 : u : : a : z ; 

hence, u = — , u 2 = —, and du = — , 

a a 4 a 

and consequently, 

/du r dz _, z 

iT^= a J?r^ = tang "T ; 

hence, by dividing by a, 



A 



dz 1 _ x z 

= — tang l — , 



a* + z* a~ * a 

the arc being estimated to the radius unity. 

225. We have seen (Art. 71), that if x represents an 
arc, and u the versed-sine, to the radius of unity, we have 

, du 

ax— . —\ 

V2u — u 2 

hence, / . = x = ver-sin~ * u + C : 

J V2u — u 2 



INTEGRAL CALCULUS. 201 

and if the arc is estimated from the beginning of the first 
quadrant, C = 0, and we shall have 

lu 



Vzm — u 2 



226. To integrate an expression of the form 
dz 



V2 



az — z 



Suppose, as before, a to be the radius of a circle, and 
we shall have (Art. 220), 

z , dz 



du = — ; 
a a 



ver-sin — 



and consequently, 

ydu r dz 

V2u-u 2 ~ J V2az-z 2 ~ « 

to the radius unity. 

Integration by Series. 

227. Every expression of the form 

Xdx, 

in which X is such a function of x, that it can be developed 
in the powers of x, may be integrated by series. 
For, let us suppose 

X = Ax a + Bx h + Caf + Dx d + &c, then, 
Xdx = Ax a dx + Bx h dx + Cx c dx + Dx d dx -f &c, 

fXdx=-^—x a * l + T ^-x b + 1 + -^- ^ + 1 4--^-V+ 1 + &c. 
J a+1 6+1 c+1 d+1 



202 ELEMENTS OF THE 

Hence, the integration by series is effected by develop- 
ing the function X in the powers of x, multiplying the 
series by dx, and then integrating the terms separately. 

Let us take, as a first example, — , 

a + x 

— dx x = dx(a + x)~\ 



a + x a -\- x 

a a z a 3 a 4 
and consequently, 

ydx C f 1 7 xdx , a?dx x z dx , e \ 
= I ( — dx — -^-H ^ — + &c.) 
a + x J \ a a* a 6 a? J 

and integrating each term separately, we obtain 



/dx 
a -\- x 



/y> /yi2 nii <yi* 

O/ lib , Ui lit « i y^» 



= log(a + a?) (Art. 218), 

G5 ~\~ X 

we have 

To determine the value of the constant, make x= 0, 
which gives 

log a = + C, or C = log a ; hence, 
log( a + *) = lo ga + !-^ + ^-£ ; +&c. > 

log( a +*)-log a =log(l+-J) = ^-|l + ^_&c. ) 
a result which agrees with the development in Art. 58. 



INTEGRAL CALCULUS. 203 

dx 



228. Let us take, for a second example 

We have, —^-5- = dx ( 1 + «T ' 5 

1 + ar 



1 + a*' 



and by developing and integrating, 

j 1+x 2 35 7 

When we make x — 0, the arc is ; hence, 

/yi<5 /y*£> •yi' 

[ -I dU UU UU o 

tang x—x 1 \- occ; 

s 3 5 7 

a result which corresponds with that of Art. 78. 

229. If, in the expression — , we place x 2 in the 

first term of the binomial, and then develop the binomial 
x 2 + 1, we obtain 

and by integrating, we have 

tang -. ;r= _l + _L__L +&c . + c. 

To find the value of the constant C, let us make the 
arc =90° = -t. This supposition will render the tan- 
gent x infinite, and consequently every term of the series 
will become 0, and the equation will give 

— tt = 0+C, or C = — *r. 
2 2 



204 ELEMENTS OF THE 

Making this substitution, we have, for the true integral. 

C dx f _! 1 11 1 ' 

/ -= = tang l x = — ie 5 4- &c. 

J x 2 +l G 2 x Sx 3 5x 5 

dx 



230. The two series, found from the expressions - 

dx l 

and -J— — , are, as they should be, essentially the same. 

For, the tangent of an arc multiplied by its cotangent, 
is equal to radius square or unity (Trig. Art. XVIII). 

Hence, if we substitute for x, in the first series, -, we 

x 

shall have, for the complemental arc, 



„-i 



111 1 



tang" 1 — = — ^ + 



'& 



x x Sx 3 5x 5 



and subtracting both members from -«■, 

— * — tang — = tanff x = — * h ir^ — ^^+ &c. 

2 5 a? to 2 x 3x 3 5x 5 

231. We have found (Art. 71), 

m" l 'x = f dx =h-x*y^dx; 
J VI — x 2 

and by developing, we find 

(l^^)"T = 1 + i_^ + ± 3 * J_ 3 ^ 6 & 

v ' 2 24246 

multiplying by <£*?, and integrating, we obtain, 

. _, , 1 x 3 , 1 3 ar 5 , 1 3 5 a: 7 , A 

232 452 4 67 ' 



INTEGRAL CALCULUS. 205 

the constant being when the arc is estimated from the 
beginning of the first quadrant. 

If we take the arc of 30°, the sine of which is equal 
to half the radius (Trig. Art. XIV), we shall have 

• -iQAo 1^111 , 131 ! ,1351 1 ,„ 

sin ^0° = - + -.-.-H — .-.-.-r+ -.-.-.-.— + &c.; 
2 23 2 J 245 2° 24 6 7 2' 

hence, 

a ■ -Ion-, rf l l ' l - l 13.1.1 1.3.5.1.1 , - \ 

*- = 6sm 1 30° = 6(-+ _ + j ; + &c), 

\2^2.3.2 3 ^2.4.5.2 5 ^2.4.6.7.2' P 

and by taking the first ten terms of the series, we find 

*• = 3.1415962, 

which is true to the last decimal figure, which should be 5. 

232. We will add a few more examples. 

dx 
1. To integrate the expression , 

V x — x 2 

By making V x = u, we have 

dx dx 2 du 



Vx — x 2 VxVl—x Vl—u 2 

But from the last series 

2du / lu 3 1 Su 5 1 3 bu 1 



i 



(u-\ \--. h-.-. h&c. ) + C; 



Vl-u 2 ^ 23 245 2467 



hence 

dx / . Ix . 1 Sx 2 .1 3 5* 3 



/ 



/, , Ix , 1 3 of 1 3 5x° , s \ .- , n 
V#-;r 2 \ 23 24 5 2467 / 

2. ^V2a^-a7 2 = (2a) 2 ^ 2 ^(l J 2 . 



206 ELEMENTS OF THE 

But 



V ~2a) ~~2~2a~~2'~^.Xa l ~ 2.4.68a 5 "" 



hence 



2 ~ 1 2 x* 112 a: 2 



Fdxy2ax — x i — — a? 2 . . — . — — -^ 

2 5 2a 2 4 7 4a 2 



113 2a? 2 



2 4 6 9 8a 3 
and consequently 

\ \ x 1 1 1 tf 2 



fdx\/2ax — x 2 = (y — ~7T' 



5 2a 2 4 7 4a 2 
_1_ j$_ _l__f 3 
2*4" 6 ' 9 8i 



— - &c. N )2a?V r 2aa: + C. 



If the radius of a circle be represented by a, and the 
origin of co-ordinates be placed in the circumference, the 
equation will be (An. Geom. Bk. Ill, Prop. I, Sch. 3), 

y 2 — 2ax — x 2 ; hence y — V 2ax — x 2 f 

and consequently (Art. 130) 

dx V 2 ax — x 2 = ydx 

is the differential of a circular segment. 

If we estimate the area from the origin, where x = 0, 
we shall have C = 0. If then we make x — a, the series 
will give the area of one quarter of the circle, if we make 
x — 2a, of the semicircle. 

n C dx \oc? 1.3a: 5 1.3.5a? 7 , 

3. / . —x 1 h&c. + C. 

J Vl+x 2 23^2.45 2.4.67 



INTEGRAL CALCULUS. 207 

a C *c ; i 1-3 1-3-5 . , _ 

4. / . =Lv = ■ -— &c. + C. 

^ Vr-1 2.2a? 2.4 At 4 2.4.6.6a: 6 T 



Integration of Differe7itial Binomials. 

234. Differential binomials may be represented under 
the general form 

x w - x dx(a + bx n f, 

in which, without affecting the generality of the expres- 
sion, m and n may be regarded as entire numbers, and n 
as positive. 

For, if m and n were fractional, and the binomial of 

the form 

1 I £ 

x 3 dx(a + bx 2 )* 

make a; = z 6 , that is, make the exponent of z the least 
common multiple of the denominators of the exponents 
of x, and we shall then have 

a: 3 dx{a + frr 2 )? = 6* 7 ck:(a + bz 3 )* f 

in which the exponents of the variable are entire. 
If n were negative, we should have, 

x m - l dx(a-\-bx- n )*, 

and by making x = — we should obtain 
z 

-z- m - l dz(a + bz n j?, 

the same form as before. 

14 



208 ELEMENTS OF THE 

Furthermore, the binomial 

may be reduced to the form 

a: » cfo(a + 6o? n ~ r )?, . 
by dividing the binomial within the parenthesis by x r , and 
multiplying the factor without by x q . 

235. Let us now determine the cases in which the 

binomial x m ~ x dx{a + bx n ) q has an exact integral. 
Make a + bx n — z q ; we shall then have 

m 

z q — a , , , „v- „ m /z? — a\ n 

* n = — j- > ( a + ^ ) 7 = z *> x = {—y- ) > 
and by differentiating, 

hence 



, • — l 



oT- 1 dx{a + bo?)' = ±z* + '- l dz(t-j-^ 



which will have an exact integral in algebraic terms when 

— is a whole number and positive (Art. 216). If — is 
n n 

negative see Art. 260. 

Hence, every differential binomial has an exact inte- 
gral, when the exponent of the variable without the paren- 
thesis augmented by unity, is exactly divisible by the 
exponent of the variable within. 

Thus, for example, the expression 

L 
x 5 dx{a-\-bx 2 y 



INTEGRAL CALCULUS. 209 

has an exact integral. For, by comparing it with the 
general binomial, we find 

m = 6, n = 2, and consequently, — = 3, 

n 

and the transformed binomial becomes 

2b \ b ) 

236. There is yet another case in which the binomial 
p_ 
x m ~ 1 dx(a-\-bx n ) < ! has an exact integral. 

If we multiply and divide the quantity within the paren- 
thesis by x n , we have 

L £ 

x m ~ l dx(a -f hx n Y = x"" 1 dx[(ax~ n -f b)x"]f 

P rtp 



= x m - l dx{ax~ n -\-b)^x 



m+—-l £. 

x q dx(ax~* -\-b)\ 



Now, if we add unity to the exponent of x without the 
parenthesis, and divide by — n, the quotient will be 

— ( h — )j and the expression will have an exact 

integral when this quotient is a whole number (Art. 235). 

Hence, every differential binomial has an exact integral, 
when the exponent of the variable without the parenthesis 
augmented by unity and divided by the exponent of the 
variable xoithin the parenthesis, plus the exponent of the 
parenthesis, is an entire number. 

237. The integration of differential binomials is effected 
by resolving them into two parts, of which one at least has 
a known integral. 

We have seen (Art. 28) that 

d{uv) — udv + vdu, 



210 ELEMENTS OF THE 

whence, by integrating, 

uv —fudv -\-fvdu, 
and, consequently, 

fudv — uv —fvdu. 

Hence, if we have a differential of the form Xdx, in 
which the function X may be decomposed into two factors 
P and Q, of which one of them, Qdx, can be integrated, 
we shall have, by making / Qdx = v and P = u, 

fPQdx = Pv-fvdP, 

in which it is only required to integrate the term fvdP. 

238. To abridge the results, let us write p for — , in 

which case p will represent a fraction, and the differential 
binomial will take the form 

x m ~ l dx(a + bx n ) p . 

If now, we multiply by the two factors x n and v~ n , the 
value will not be affected, and we obtain 

x m - n x n ~ 1 dx(a + bx n ) p . 

Now, the factor x n ~ l dx(a -f bx n ) p is integrable, whatever 
be the value of p (Art. 217) ; and representing this factor 
by dv, we have 

(p+ l)nb 

and, consequently, 

fx m ~ l dx(a 4- for")* — 

/ x r -7 r-T- fcc m - n ~ l dx(a -f kr n ) p+l . 



INTEGRAL CALCULUS. 211 

But, fx m - n ~ x dx{a + bx n ) p+i = 

fx m - n - l dx(a + bx n ) p (a + bx n ) = 
afx m ~ n - l dx{a + fca?")* + bfx m ~ x dx{a + ta")" ; 

substituting this last value in the preceding equation, and 
collecting the terms containing the integral 

fx m - l dx(a+bx n ) p , 
we have 

( 1 +7=^V*-- , *<« + **■>' = 

x m ~ n (a + kc n ) p + l - a(m - n) f x m - n ~ x dx{a + &£) p . 
~~ (p+l)n& ; 

whence, 

formula (A.) fx m ~ l dx{a + bx n ) p = 

x m - n {a + bx n ) p + l -a( m — n) f x m - n - l dx(a + bx n ) p 
b(pn + m) 

This formula reduces the differential binomial 

fx m - 1 dx(a + ta?T to that of Jx m - n ~ l dx{a + bx n ) p ; 

and by a similar process we should find 

fx m - n - l dx{a + bx n ) p to depend on fx m ~* n - l dx(a + bx n f; 

and consequently, each process diminishes the exponent 
of the variable without the parenthesis by the exponent 
of the variable within. 

After the second integration, the factor m — n, of the 
second term, will become m — 2n; and after the third, 
m — 3n, &c. If m is a multiple of n, the factor m — n, 
m — 2n, m — Sn, &c, will finally become equal to 0, and 
then the differential into which it is multiplied will disap- 



212 ELEMENTS OF THE 

pear, and the given differential will have an exact integral, 
which corresponds with the result of Art. 235. 

239. Let us now determine a formula for diminishing 
the exponent of the* parenthesis. 
We have 

fx m - l dx(a + bx n ) p = foc m - l dx(a 4 bx n ) p ~ l {a + bx n ) = 
afx m - l dx(a + bx n ) p ~ l + bfx m+n ~ l dx{a + bx n ) p ~\ 

Applying formula (A) to the second term, by placing 
m 4- n for m, and p — 1 for p, we have 

fx m + n - l dx(a-\-bx n ) p - 1 = 

x m (a -f bx n ) p — amfx m ~ 1 dx(a 4- bx n ) p ~ l 
b(pn + m) 

Substituting this value in the last equation, we have 

formula (B) Jx m ~ l dx{a + bx n ) p = 

x m (a + bx n ) p -\-p?iafx m - 1 dx(a 4- bx n ) p ~ l 
pn-\-m 

which diminishes the exponent of the parenthesis by unity 
for each integration. 

240. By means of formulas (A) and (B), we reduce 

fx m - x dx(a + bx n ) p to fx m ~ rn - x dx{a + bx n ) p - ; 

rn being the greatest multiple of n which can be taken 
from m— 1, and s the greatest whole number which can 
be subtracted from p. 

For example, fx 7 dx(a 4- bx 3 ) 2 is reduced, by formula 
(A), to 

fx*dx{a + bx 3 ) 2 , and then to / xdx (a 4- bx 3 ) 2 : 



INTEGRAL CALCULUS. 213 

5 

and by formula (B) fxdx(a + bx 3 )' 2 , reduces to 

1 1 

fxdx(a + bx 3 )' 2 , and finally to fxdx(a -f- bx 3 y. 

241. It is evident that formulas (A) and (B) will only 
diminish the exponents m — 1 and p, when in and p are 
positive. We will now determine two formulas for dimin- 
ishing these exponents when they are negative. 

We find from formula (A) 

Jx m - n - x dx(a + bx n ) p = 

x m ~"(a + bx n ) p+x - b{ m + np) f x m ~ l dx{a + bx») p m 
a(m — n) 

and placing for m, — m + n, we have 

formula (C) ■fx" m ' l dx{a-\- bx»Y = 

x~ m (a + &y n ) ?+1 .+ b(m-n- np) fx- m + n - l dx(a + for")* 

— a?ra 

in which formula, it should be remembered that the nega- 
tive sign has been attributed to the exponent m. 

242. To find the formula for diminishing the exponent 
of the parenthesis when it is negative. 

We find, from formula (B), 

fx m - l dx(a + bx n ) p - l =: 

x m (a + bx n ) p — (m + np)fx m - 1 dx(a + bx n ) p 
pna 

writing for p, — p + 1, we have 

formula (D) Jx m ~ l dx{a + bx»)- p = 

x m (a + bx n )~ p + l -(m + n- np)Jx m - l dx (a + bx n )~ p+l 
(p — \)na 



214 ELEMENTS OF THE 

This formula does not apply to the case in which p = 1. 
Under this supposition, the second member becomes infi- 
nite, and the differential becomes that of a transcendental 
function. 

243. It is sometimes convenient to leave the variable in 
both terms of the binomial. We shall therefore determine 
a particular formula for integrating the binomial 

a* (2ax - 'a*)~*dx = -y= === : 

V 2 ax — x 2 

This binomial may be placed under the form 

-.L J. 

fx * dx(2a — x) 2 , 

and if we apply formula (A), after making 

m = q+— , w = i, p= — — , a = 2a, b = - 1, 

we shall have 

fx 2 dx(2a—x) 2 = 

and if we observe that 

.,-1 j-i 1 9 -l 9 -i -1 
x 2 =07 a? 2 a? 2 =x x 2 , 

and pass the fractional powers of a? within the parentheses, 
we shall have 

x q dx 



formula (E) / 



V2ax — x 2 
x q -W2ax^a? (2q-l )a C x q ~ x dx 

q q J V2ax-x 2 > 



INTEGRAL CALCULUS. 215 

which diminishes the exponent of the variable without the 
parenthesis by unity. If q is a positive and entire num- 
ber, we shall have, after q reductions 

y* dx x 

7 - = ver-sin" 1 — . (Art. 226). 
VZax-x 2 a K 



Integration of Rational Fractions, 

244. Every rational fraction may be written under the 
form 

P x n ~ l + Q x n ~~ + R x + S d 

P'x n +QV 1 - 1 +R'x+S' *' 

in which the exponent of the highest power of the varia- 
ble in the numerator, is less by unity than in the denomi- 
nator. For, if the greatest exponent in the numerator was 
equal to or exceeded the greatest exponent in the denomi- 
nator, the division might be made, giving one or more 
entire terms for a quotient and a remainder, in which the 
exponent of the leading letter would be less by at least 
unity, than the exponent of the leading letter in the divisor. 
The entire terms could then be integrated, and there 
would remain the fraction under the above form. 

Place the denominator of the fraction equal to : that 
is, make 

P'x n + QV- 1 R'x+S' = 0, 

and let us also suppose that we have found the n binomial 
factors into which it may be resolved (Alg. Art. 264). 
These factors will be of the form x — a, x — b, x — c 
x — d, &c. Now there are three cases : 



216 ELEMENTS OF THE 

1st. When the roots of the equation are real and 
unequal. 

2d. When they are real and equal. 
3d. When there are imaginary factors. 
We will consider these cases in succession. 

1st. When the roots are real and unequal. 

adx 



245. Let us take, as a first example, 



By decomposing the denominator into its factors, we 
have 

adx adx 

x 2 — a 2 (x — a) (x + a) 9 

and we may make 

adx —(A B \ 7 

(x — a)(x-\-a) \x — a x + a) 

in which A and B are constants, whose values are yet to 
be determined. In order to determine these constants, 
let us reduce the terms of the second member of the 
equation to a common denominator ; we shall then have 

adx (Ax + Aa + Bx — Ba) dx 



(x — a)(x-\-a) (x — a)(x + a) 

In comparing the two members of the equation, we find 

a = Ax + Aa + Bx — Ba ; 
or, by arranging with reference to x, 

(A + B)x-{-(A-B-l)a = 0. 
But, since this equation is true for all values of x, the 



INTEGRAL CALCULUS. 217 

coefficients must be separately equal to (Alg. Art. 208) : 
hence 

A + B = 0, and {A- B- l)a = 0, 
which gives 

A=\ B=-K 

2' 2' 

Substituting these values for A and 5, we obtain 

adx jdx -_dx 
x 2 — a 2 x — a x-{- a 1 

and integrating, we find (Art. 218) 

fjhr^ = \ lo ^ x ~ a ) ~ y 1o s(^ + fl ) + c > 

and, consequently, 

.^ T i i i « 3 + bx 2 7 

246. Let us take, as a second example, — ~dx. 

a^x — ar 

The factors of the denominator are x and a 2 — x 2 ; but 
« 2 — x 2 = (a -{• x) (a — x) : 



hence, the given fraction becomes 
a 3 + bx 2 



x(a — x)(a-rx) 
Let us now make 



dx. 



at + bx 2 A B C 



x(a — x){a + x) x a — x a + x* 



218 ELEMENTS OF THE 

reducing the terms of the second member to a common 
denominator, we have 

a" + bx 2 Aa 2 - Ax 2 -\- Bax+Bx 2 + Cax -Cx 2 



x(a — x) (a + oc) x(a — x)(a -\- x) 

and, comparing the like powers of x (Alg. Art. 208), 

B-A-C = b, Ba+Ca = 0, Aa 2 = a 3 . 
From these equations, we find 

a + b a + b 
A = a. B — , C= , 

2 2 

and substituting these values, we obtain 

a 3 + bx 2 , dx a -\-b , a + b , 

— 5 -ax == a 1 -ax -ax ; 

a z x — x 3 x 2(a — x) 2(a -\- x) 

and integrating (Art. 218), 

fa 3 -\-bx 2 , . d + b, . . 

/ —5 -gw? = aloga? loff(a — #) 

J a 2 ^-K 3 ° 2 & ' 

~^T log(a + ^)+C 
= alog# — °—- — [log(a — a?) + log(a + x)] + C 

= alogx l°g(ft — x ) i a + a?) + C 

= aloga: log(a 2 — a? 2 ) -f- C 



= aloga? — (a + 6)log -/a 2 — x 2 + C. 

3^p 5 

247. Let us take, for a third example, — dx. 

xr — 6x + 8 



INTEGRAL CALCULUS. 219 

Resolving the denominator into the two binomial factors 
(Alg. Art. 142), 0-2), 0-4), we have 

3x-o A B 

-z = (- , hence 

x 2 -6x + S x — 2 a: -4 

3a:— 5 _ Ax — 4:A + Bx — 2B 
#* -6a? + 8 ~ x 2 -6x + 8 

and by comparing the coefficients of a?, we have 

-5= -4A-2£, 3 = A + B, 

which gives 

2 2 

and substituting these values, we have 

f 3*-5 &= _irA + l/-A +c 

*/ a? 2 -6a? + 8 2 J x — 2 2 J x— 4 

- ylogO - 4) - ^logO - 2) + C. 

248. Let us take, as a last example, 

a?6?a? 

x 2 + 4 aa? — b 2 ' 

Resolving the equation 

x 2 + 4ax-b 2 = 0, 
we find 

x = - 2a + VT^+I?, a: = - 2a - \/4a 2 + 6 2 , 
and consequently, for the product of the factors, 
(a? + 2a + V4a 2 + 6 2 ) (a?+2a- V^<?+?) = a?+4<H7--#. 



220 ELEMENTS OF THE 

To simplify the work, represent the roots by — K and 
— L, and the factors will then be 

x + K, x + L, 

and we shall have 

x A . B 

hence 



x 2 + 4 ax — b 2 x -f- K x + L 

x _Ax + AL + Bx+BK 

x> + 4:ax-b 2 ~ x 2 + 4ax-'b 2 ' 

whence, 

AL + BK = 0, A + B=l, 
and, consequently, 

hence, 

249. In general, to integrate a rational fraction of the 
form 

Px m ~ l + Qx m - 2 +R X +S d 

x m +Q'x m - 1 +R'x+S' 

1st. Resolve the fraction into m partial fractions, of 
which the numerators shall be constants, and the denomi- 
nators factors of the denominator of the given fraction. 

2d. Find the values of the numerators of the partial 
fractions, and multiply each by dx. 



INTEGRAL CALCULUS. 221 

3d. Integrate each partial fraction separately, and the 
siun of the integrals thus found ivill be the integral 
sought. 

250. The method which has just been explained, will 
require some modification when any of the roots of the 
denominator are equal to each other. When the roots are 
unequal, the fraction may be placed under the form 

Px'+Qx' + Rof+Sx+T 



(x — a) (x — b) (x — c) (x — d) (x — e) 
x — a x — o x — c x — a x — e 



if several of these roots are equal, as for example, 
a = b = c, the last equation will become 

Px* + Qx 3 + &c. _• A + B + C D E 



(x — af (x — d) (x — e) x — a x — d 

in which A + B -f C may be represented by a single con- 
stant A! . 

Now, in reducing the second member of the eouation to 
a common denominator with the first, and comparing the 
coefficients of the like powers of x, we shall have five 
equations of condition between three arbitrary constants, 
A', D, and E : hence, these equations will be incompati- 
ble with each other (Alg. Art. 103). 

If, however, instead of adding the three partial fractions 

ABC 



x — a? x — o' x — a? 



which have the same denominator, we go through the 



222 ELEMENTS OF THE 

process of reducing them to one, their sum may be placed 
under the form 

A! + B'x + Cx 2 
{x - af 

or, by omitting the accents, 

A + Bx + Cx 2 
(x-af ' 

251. Let us now make 

x — a = z, and consequently, x = z -\- a ; 
we shall then have 

A + Bx + Cx 2 A + Ba + Ca 2 + Bz + 2Caz 4- Cz l 



(x - af 



A-\-Ba+Ca 2 B + 2Ca C 
-?; "i Is r —- ; 



substituting for z its value, and representing the numera- 
tors by single constants, we have 

A + Bx + Cx 2 _ A' B' O 

(x — a) 3 ~ (x — af (x— af x — a y 

the form under which the fraction may be written. 

Since the same reasoning will apply to the case in 
which there are m equal factors, we conclude that 

Px m ~ l + Qx m ~* -\-Rx-\-S __ 

(x - af' 
A A f A"_ A* . / 

-l "T" /„ „\m-2 • • •"• T 



{x — a) m {x — a) m ~ l {x — a) m ~' z x—a 

252. In order, therefore, to integrate the fraction 
Px 4 + Q'x 3 + &c 



(x — af (x — d) (x — e) 



dx, 



INTEGRAL CALCULUS. 223 



place it equal to 



A A' A" D , E 

s+— - + -— i + — 



(x — af (x — a) 2 x — a x — 

then, reducing to a common denominator, and comparing 
the coefficients of the like powers of x, we find the values 
of the numerators of the partial fractions. Multiplying 
each by dx, and the given fraction may be written under 
the form 

A j A' , A! 1 , D ' , E . 

; dx 4- -, T5 "^ + 7 : a# H 7 dx -\ ax. 



(x—a) 3 (x — a) 2 (x—a) x — d x — e 

The first two fractions may be integrated by the method 
of Art. 217, and the three last by logarithms. Hence, finally, 



s 



Pa ? 4 4- Qx 3 + Rx 2 + &c+ T d _ A A f 

(x — a) 3 (x — d)(x — e) 2(x — a) 2 x — a 

+ A" log (a? — a) + DlogO — d) + ElogO - e) + C. 

253. Let it be required to integrate the fraction 
2 ax 



(oc 4- af 
We have 



S-C&P. 



2aa? A A! 



(x -\- a) 2 (x + a) 2 x + a' 

reducing the fractions of the second member to a common 
denominator, and comparing the coefficients of x in the 
two members, we have 

2a = A f and A + Ala — : 

hence, 

A=-2a 2 . and A f = 2a; 
15 



224 ELEMENTS OF THE 

and, consequently, 

2axdx 2a 2 dx 2adx 



(x -f a) 2 " (x -f a) 2 (a? + a) ' 
hence, (Arts. 217 & 218), 

r 2axdx 2 a 2 



I 



(x 2 -f a) 2 a?+a 
254. Let us find the integral of 
x?dx 



2 a log (x -f a). 



a? 3 — aar — a 2 a? 



By placing the denominator equal to 0, we see that, by 
making x = a, the terms will destroy each other : hence, a 
is a root of the equation, and x—a a factor. Dividing by 
x — a, the quotient is x 2 — a 2 : hence, the fraction may be 
placed under the form 

x?dx _ x 2 dx 

(x 2 — a 2 ) (x — a) (x + a) (a? — a) (x — a) 

(x — a) 2 {x -f a) ' 
Let us now make 

a? A A f B 



(x — a) 2 (x + a) (x—ay (x — a) a? -fa 

Reducing the terms of the second member to a common 
denominator, we have 

x 2 A {x -f a) + A' [x 2 - a 2 ) + 5 (a; - a) 2 



(a? — a) 2 (a? + a) (a? — a) 2 (a? + a) 

and developing, and comparing the coefficients of the like 



INTEGRAL CALCULUS. 225 

powers of x, we obtain the equations 

A f + B=l, A-2Ba = 0, Aa- A' a 2 + Ba 2 = 0. 

Multiplying the first equation by a 2 , and adding it to the 
third, we have 

Aa + 2Ba 2 = a 2 ; 

then multiplying the second by a, and adding it to the last, 
we have 

a 2 = 2Aa, and consequently, A= — a ; 
substituting this value of A, we find 

5 = -i and A'--?-. 

4 4 

Substituting these values of A, A f , and B, we have 
a?dx adx Sdx dx 



(x — a) 2 (x + a) 2(x — af 4(x — a) 4(a? + a)' 
and consequently, 

x i -a^-a 2 x + a*~~ 2{x-ay± g[X ~ a) 
+ _Llog(tf-M)+C. 

255. We can integrate, in a similar manner, when the 
denominator contains sets of equal roots. Let us take, as 
an example, 

adx adx 

( X 2 -l) 2 = (x-\) 2 (x+l) 2 ' 



226 ELEMENTS OF THE 

Make 



,-^- + *^+ B ' 



(x-\f{x+\f {x-l) 2 x-l (a?+l) 2 a?+l 

reducing the second member to a common denominator, 
we find the numerator equal to 



2- 



A{x+\J + A!{x-\\x+\) 2 + B{x-Yf + B\xAr\){x-\) 

and comparing the coefficients with those of the numera- 
tor of the first member, we have the following equations : 

A 1 + B' = 0, 
A +A'+ B-B' = 0, 
2A -A f -2B-B' = 0, 
A -A f + B + B f = a. 

Combining the first and third equations, we find A = B; 
and combining the second and fourth, gives 2 A + 2B = a: 
hence, we have 

A-B-— A'--— B'- — ■ 

consequently, the given differential becomes 

\ V~ dx dx dx dx ~] 

~± a \_(x-\y + (^+i) 2 ~ x~^~\ + ^+tJ' 

and by integrating, 

/(^r ? =K-i-r+-T- ,0 ^- 1 )+ l0 ^+ 1 )]+ c - 

256. If an equation of the second degree has imaginary 
roots, the quantity under the radical sign will be essentially 



INTEGRAL CALCULUS. 227 

negative (Alg. Art. 144), and the roots will be of the form 

X — +C + 5 -y/— 1, cT = =F a — b -y/ — 1 , 

and the two binomial factors corresponding to the roots 
will be 



(x ± a —6 V - 1 ) O ± a -h 6 V^ 17 ^) = a- 2 ± 2a# + a 2 + & 2 . 
Hence, for each set of imaginary roots which arise from 
placing the denominator of the fraction equal to 0, there 
will be a factor of the second degree of the form 

x 2 ± 2ax + a 2 + b 2 . 
257. If the imaginary roots are equal, we shall have, 



a = 0, x = + b y— T , a? = — 6 -y/— 1, 

and the factor will become x 2 + b 2 . 
In the equation, 

a? — 6cx+ 10c 2 = 0, 
the roots are, 



# = 30 + 0^—1, a? = 3c — cV— 1; 

comparing these values of x with the general form, we 
have 

a = — 3c 6 = c, 

and the given equation takes the form 

x 2 - 6cx + 9c 2 + c 2 = 0. 

Comparing the roots of the equation, 

#2-1-40?+ 12 = 0, 

with the values of x in the general form, we have 

a - 2, b = </§, 



228 ELEMENTS OF THE 

and the equation may be written under the form 
a? + ±x + 4 + 8 = 0. 

258. Let us first consider the case in which the deno- 
minator of the fraction to be integrated contains but one 
set of imaginary roots. The fraction will then be of the 
form, 

P+Qx + Rx 2 +Sx 3 + &c. 



(x — a) [x — b) [x-h){x 2 +2ax + a 2 +b 2 ) 

which may be placed under the form 



dx, 



Adx Bdx , Hdx , Mx + N , 



x — a x — b ' x — h x 2 + 2ax + a 2 + b 2 

The first three fractions may be integrated by the methods 
already explained : it therefore only remains to integrate 
the last, which may be written under the form 

Mx + N 7 
•Ax. 



(x + a) 2 + b 2 
If we make x + a = z, the expression becomes 

Mz + N— Ma A 
?"+P ' 

and making N — Ma = P, it reduces to 

Mz+ P 



z 2 + b 2 



dZy 



which may be divided into the parts, 

Mzdz Pdz 



which may be integrated separately. 



INTEGRAL CALCULUS. ggg 

To integrate the first term, we have 

fMzdz _ f zdz M f 2zdz 
J J + ^ ~ M J z~ + b~~ 2J z 2 + 6 2 ' 

in which the numerator, 2zdz, is equal to the differential 
of the denominator: hence (Art. 218), 

fMzdz M, , , , m 

or by substituting for z its value, a? -f a, 

CMzdz M, r/ x2 . , 2 , 

= — log(a ? 2 +2^ + a 2 + & 2 ) 
-4 



= Mlog Va* + 2aa? + a a + #. 
Integrating the second term by Art. 224, gives 
f Pdz p- _ x fZ\ 

or by substituting for z its value, x + a, and for P, 
iV — Ma, we have 

/Pdz N—Ma _i/a?4-a\ 
?+£ = — 6— tang (—) ; 

and finally, 

f Mx + N 
J a? + 2ax + a 2 + b 2 

M\og^x 2 +2ax + a 2 + b 2 + ^— tang" 1 (^^p). 

259. Let us take, as an example, the fraction 
c+fx , 



230 ELEMENTS OF THE 

in which, if +1 be substituted for x, the denominator 
will reduce to : hence, x — 1 is a factor of the denomi- 
nator. Dividing by this factor, the fraction may be put 
under the form 

c+fx 



( x -l)(x 2 + x+l) 



dx, 



in which x 2 -f- x -f 1 is the product of the imaginary 
factors. Placing this product equal to 0, finding the roots 
of the equation, and comparing them with the general 
values in the form 

x 2 +2ax + a 2 + b 2 =0 } 
we find 

1 , /T 

a— — b = \/ — . : 

2 V 4 

We may place the given fraction under the form 

c+fx _ A Mx + N m 

{X—I) (^ + 07+ l)~ X- 1 X? + X+ 1 ' 

reducing the second member to a common denominator, 
and comparing the coefficients of x in the numerator with 
those of x in the numerator of the first member, we obtain 

3 3 3 

Substituting these values of M and N, as also those of a 
and b, in the general formula of Art. 258, and recollecting 
that 

ru*_ cjrf_ rj*_ = c+f l{x _ , } 

J x-l 3 J x-\ 3 oV ; ' 



INTEGRAL CALCULUS. 231 

we find 



./ 



— 1 o o 



f-c 



+ —f= tan § ' 

V 3 



•±il+c. 



260. The equation which arises from placing the de- 
nominator of the fraction equal to 0. may give several 
pairs of imaginary roots respectively equal to each other. 
In this case, the factor x 2 ± 2 ax + a 2 -f b 2 will enter 
several times into the denominator, or will take the form 

(x 2 +2ax + a 2 +b 2 ) p ; 

and hence, that part of the fraction which contains the 
pairs i)i equal and imaginary roots, must be placed under 
the form (Art. 251) 

if+ Kx H' + K'x 

(x 2 + 2ax + a 2 + 6 2 ) p (a 2 + 2aa? + a 2 + b 2 ) p ~ l 
H" + #"* H" + K n x 



+ 



(a7 2 4-2^ + a 2 -f^ 2 ) p " a ^ 2 + 2a* 4- a 2 4- 6 2 

Now, reducing to a common denominator, and comparing 
the coefficients, we find the values of the constants 

H, K, H, K', H" f K" H n , K n . . . 

after which, multiply each term by dx, and then integrate 
the terms separately. 

Since all the terms are of the same general form, it will 
only be necessary to integrate the first term, which may 
be written under the form 

H +Kx 



[(x + a) 2 + V 



V dx > 



232 ELEMENTS OF THE 

which, if we make x + a = z, will reduce to 
H-Ka + Kz . 

and making M= H— Ka, it will become 

M+Kz , .BMz , Mdz 

c/z = — — — az + 



(6 2 + z 2 )* (6 2 + z 2 ) p ' (6 2 -f z 2 )" 

The first term of the second member may be placed under 
the form 

Kf{b 2 + z 2 )~ p zdz, 

and integrating by the formula of Art. 217, we have 
Kzdz 1 K 1 



/■ 



i+C. 



(6 2 +z 2 ) p 2 (i-p)(6 2 +z 2 r 

It then only remains to integrate the second term 
Mdz 

By comparing the second member of this equation with 
formula (D), Art. 242, we see that it will become identical 
with the first member of that formula, by supposing 

?7i = 1 , a = b 2 , 6=1, and n = 2 ; 

and hence, by means of that formula, the exponent — p 
may be successively diminished by unity until it becomes 
— 1, when the integration of the term will depend on 
that of 

b 2 + z 2 ' 
But we have already found (Art. 224), 

C dz 2 1 _ 4 /*\ 



INTEGRAL CALCULUS. 233 

and hence the fraction may be considered as entirely in- 
tegrated. 

261. It follows, from the preceding discussion, that the 
integration of all rational fractions depends on the follow- 
ing forms : 

1st. fx m dx = - . 

J m+1 

2d - f^ ±l0 ^ a±xy - 

3d- /V^ = -tang-(*\ 

J a 1 -\-x i a ° \ a J 

Integration of Irrational Fractions. 

262. The method of integrating rational fractions having 
been explained, we may consider an irrational fraction as 
admitting of integration when it is reduced to a rational 
form. 

263. Every irrational fraction in which the radical 
quantities are monomials, may be reduced to a rational 
form. 

Let us take, as an example, 



V x — \a 



— dx, or 



%fx— Vic 

Having found the least common multiple of the indices 
of the roots, (which indices are the denominators of the 
fractional exponents,) substitute for x a new variable, z, 
with this common multiple for an exponent, and the frac- 
tion will then become rational in terms of z. 



234 ELEMENTS OF THE 

In the example given, the least common multiple is 6 ; 
hence we have 

x = z 6 and y'x = z 2 , y'x = z 2 , dx = 6z*dz ; 
and substituting these values, we obtain 

yx — yx z — z \ — z 

an expression which may be integrated by rational frac- 
tions ; after which we may substitute for z its value, y'x. 

264. If the quantity under the radical sign is a polyno- 
mial, the fraction cannot, in general, be reduced to a 
rational form. We can, however, reduce to a rational 
form every expression of the form 



X{VA + Bx± Cx 2 )dx, 

in which X is supposed to be a rational function of x. 

If we write a denominator 1, and then multiply the 
numerator and denominator by Va + Bx ± Cx 2 , the 
expression will take the form 

X'dx 



VA + Bx+Cx 2 ' 

in which X f is a rational function of x: hence the two 
forms are essentially the same. 

If now, we can find rational values for Va+ Bx i Cx 2 
and for dx, in terms of a new variable, the expression will 
take a rational form. 

There are two cases to be considered: 1st., when the 
coefficient of x 2 is positive ; and, 2d, when it is negative. 



INTEGRAL CALCULUS. 235 

Let us consider them separately. First, make 



VA + Bx + Cx 2 = VC \/~ + ^roc + x 2 

G G 

= V~C V a + bx + x 2 , 

in which a = — , b = — . 
G G 

In order to find rational values for dx and Va + k+^, 

place 

Va + bx + a? = x + z, (1 ) 

from which, by squaring both members, we find 

a + bx = 2xz + z 2 , (2) 

and hence, 

z 2 — a , nS 

*=b=rs (3) 

and substituting this value in equation (1), 

V« + bx + x 2 = h^; 

6 — 2z 

and by reducing to the same denominator, 

/ i 5 z 2 — bz + « , . N 

V« + &tf + a? = — -H-. (4) 

6 — 2z 

Let us now find the value of dx in terms of z. For this 
purpose we will differentiate equation (2), we then find 

bdx = 2xdz + 2zdct + 2zdz ; 

whence we have 

(6 — 2z)dx = 2 (a? + ^r)(fe ; 



236 ELEMENTS OF THE 

and by subtracting equations (1) and (4), and substituting 
for x + z the value thus found, we have 

,, o \ 7 2{z 2 — bz -\-a) , 
(b - 2z)dx = - -±— Ldz, 

a — 2z 

, , 2(z 2 -bz + a) . ,_. 

and dx=- {b _ 2z) 2 dz - ( 5 ) 

265. Let us take, as an example, 
dx 



xVa + Bx+Cx 2 ' 
which may be written under the form 

dx 



V C x x Va + bx + x 2 



and substituting the values of Va -\-bx-\- x 2 and dx, from 
equations (4) and (5), we have 

dx 2dz 



Va + bx + x 2 b-2z' 

and multiplying the denominator by the value of x, in 
equation (3), 

dx 2 dz 

xVa + bx + x 2 ~ * 2 -a ' 

and then by Vc, we have 

dx dx 2dz 

■-, or 



VC x x Va + bx + x 2 x VA+Bx+Cx* (z 2 -a) VC 

which is a rational form, and may be integrated by the 
methods already explained. 







INTEGRAL CALCULUS. 


266, 


, Let us 


take, c 


is a second example, 
dx 




Vh + c z x 2 


which 


may be 


placed 


under the form 
dx 




V?+'' 



237 



and comparing this with the form of Art. 264, gives 

c = VC, b = 0, -g = a. 

Hence, 

cfo? 1 r dx 



/dx 1 r a 



Having placed 

-y/a + x 2 — z + x, 
we found, Art. 264, equations (5) and (4), 

dcc=--^dz i ^ /a + ^ = Z ~2^ : y 
hence 

/dx r dz , 

. = I = — logz. 

Va + x 2 J z 

Substituting for z its value, and multiplying by — , we 

c 

have 

and substituting for a its value, — , we have 



238 ELEMENTS OF THE 



f 



dx 1 , 

— = = log 

Vh + c¥ c 



— ( -yjh -f- cPx 2 — ca?) 



+ C 



log log( V/* + c 2 ar — cx) + C. 



But since the difference of the squares of the two terms 
within the parenthesis is equal to h, it follows that if h 
be divided by the difference of the terms, the quotient will 
be their sum (Alg. Art. 59). But the division may be 
effected by subtracting their logarithms. Let us, then, 
add to, and subtract from, the second member of the equa- 
tion, — log/*. We shall then have, 



/ 



= log log A-J — log h log( VA-f cV— cx)-\- C j 

V^+cV c c c c c 

or by representing the three constants — log log/?, 

c c c 

and C, by a single letter C, we have 



s- 



dx 



= J_log( V/i + cV + ex) + C. 



Vft+cV c 
267. Let us take, as a third example, 



dx^/m 2 + o?. 
Comparing this with the general form, we find 
a — rn? and 6 = 0; 
hence (Art. 264), 

-, * 2 +™ 2 mJ ,__ (* 2 + ™ 2 ) 



V??? + x 2 = and cfo = — - — — - -'- dz ; 

2z Zz 2 



INTEGRAL CALCULUS. 239 

and consequently. 



2 I ™2\2 



(z 2 + 7/r) 



d x Vnr+a*=- ' ^ } dz, 

4:Z J 

which is rational in z ; and, having found the integral in z, 
substitute the value of z in terms of x. 

268. Let us now consider the case in which the coeffi- 
cient of x 2 is negative. We have 



Y A + Bx - Cx 2 = VC yl 



# 2 



= VC V<2 -{-bx — x 2 . 
Jf now, we make as before, 



Y a + bx — x 2 = x-{-z, 

and square both members, the second powers of x in each 
member will not cancel, as before ; and therefore, x can- 
not be expressed rationally in terms of z. We must, 
therefore, place the value of the radical under another 
form. We will remark, in the first place, that the bino- 
mial a + bx — x 2 , may be decomposed into two rational 
factors of the first degree, with respect to x. For, if we 

make 

x 2 — bx — a = 0, 

and designate the roots of the equation by « and *', we 
have (Alg. Art. 142) 

(x 2 — bx — a) = (x — «) (x — a!\ 

and consequently, by changing the signs, 

(a + bx — x 2 ) — — (x — «) (x — *') =z(x — cc) (*' _ x ) f 
16 



240 ELEMENTS OF THE 

and placing the second member under the radical, we 
may make 



V(x-*)(x'-x)=--{x-*)z; (1) 

squaring both members 

(x — *) {of — x) = (x — ct) 2 Z 2 , 

and by suppressing the common factor x — *, 

et f - x = (x-«)z 2 , (2) 
whence, 







X- 


*' + *z 2 




and 


X- 


— at,— 


«' + *z 2 
\+z 2 


- cc 


or by reducing, 












.r — 


- ct — 


at! — ct 


rai 



which, being substituted in the second member of equa- 
tion (1), gives 



V(«— )K-*) = T - j - ? *; (4) 
and by differentiating equation (3), we obtain 

^~(iW^' (5) 

269. To apply this method to a particular example of 
the form 

dx 
Va -i-bx — x 2 



INTEGRAL CALCULUS. 241 



substitute the values of V a + bx — or and dx, found i r ■ 
equations (4") and (5) : we find 

dx 2(*' — *)z _ 2dz 



Va + bx-ar {l+z j z ^l > + * 



hence 

J Va + bx — x 2 
or, by substituting for z its value from equation (1), 



dx . „, 

- — 2tang _1 z + C; 






= C-2tan 






270. If, in the last formula, we make 

a—\ and 6 = 0, 

the trinomial under the radical will become 1 — x 2 , and 
the roots of the equation x 2 — 1 = are 

*= — 1 and eef = 1. 

Substituting these values, and the general formula becomes 

and if we suppose the integral to be when x = 0, we 
shall have 

0=C-2tang" 1 (l) 
= C - 2(45°) (Trig. Art. VIII) 

= C - 90° : hence C = — . 

2 



242 ELEMENTS OF THE 

Substituting this value, and we have 

/dx v _ x f\—x 
— 2tan£ \ . 



Vl 
271. We have already seen (Art. 219) that 
dx 

and hence, 



/dx 



sin l x; 



V 1+07 



2 tang" 



should also represent the arc of which x is the sine 
To prove this, we have (Trig. Art. XXV) 

n . 2 tang A 



1 — tang 2 A 

/\ x 

Substituting for tang A, \J , and reducing, we have 

V 1 — (- x 

tang2A = ^IE2; 



/l — X 

that is, twice the arc whose tangent is W is equal 

▼ 1 "T~ X 
Vl — a? 2 

to the arc whose tangent is .. 

x 

But the arc whose tangent is ~ , is the com- 



plement of the arc whose tangent is . , (Trig. 

V l — x 2 

Art. XVIII) ; and this arc has x for its sine. Hence, 

either member of the equation 



INTEGRAL CALCULUS. 24,3 



/ ,. = 2 tang -1 W 

represents the arc whose sign is x. 

272. Let us take, as a last example, the differential 



dx v2 ax — x 2 . 

In comparing this with the general form, we find (Art 
268) 

* = and x f = 2a ; 



and Art. 268, equations (4) and (5), give 



■\/x(2a — x) — -, dx 



2az , 4 az 



+ z 2 ' (1 + z 2 ) 2 

Substituting these values, we have 

8a 2 z 2 dz 



dz. 



dx V 2 ax — x 2 



(1 + * 2 ) 



2\3 ' 



which may be integrated by the method of rational 
fractions. 

Rectification of Plane Curves. 

273. The rectification of a curve is the expression for 
its length. When this expression can be found in a finite 
number of algebraic terms, the curve is said to be rectifiable, 
and its length may be represented by a straight line. 

274. The differential of the arc of a curve, referred to 
rectangular co-ordinates, is (Art. 128) 

dz = -y/ dx 2 -f dy 2 . 



244 ELEMENTS OF THE 

Hence, if it be required to rectify a curve, given by its 
equation, 

1st. Differentiate the equation of the curve. 

2d. Combine the differential equation thus found with 
the given equation, and find the value of dx 2 or dy 2 in 
terms of the other variable and its differential. 

3d. Substitute the value thus found in the differential 
of the arc, which will then involve but one variable and 
its differential. Then, by integrating, we shall find an 
expression for the length of the arc, estimated from a 
given point, in terms of one of the co-ordinates. 

275. Let us take, as a first example, the common para- 
bola, of which the equation is 

y 2 2= 2px. 

Differentiating, and dividing by 2, we have 

ydy =pdx, 

and consequently, 

V 

substituting this value in the differential of the arc, we 
have 



dz^yjdtf + ^dy* 



V 

= —dyVp 2 + y 2 ; 
P 

which, being integrated by formula (B) Art. 239, gives, 

by supposing m = 1, a =p 2 , 6=1, n = 2, p = — , 

lit 



INTEGRAL CALCULUS. 245 



J y * * 2 2 J Vf + y 2 

and integrating the second term by the formula of Art. 
266, we have, after maiung h—p 2 , c 2 —l, 

f-i=== = lo S ( . VfTf + y) ; 

J \p 2 + y 2 
and consequently, 

If we estimate the arc from the vertex of the parabola, 
we shall have 

y — for z — : hence 

= |-logp+C or C=--|lo gi >; 

and consequently, 



„ vy + y 2 , p_, fVf_±f_±T\ . 
2v 2 S \ p J 9 



2p ' 2"°\ p 

and hence, the value of the arc, for a given ordinate y, can 
only be found approximatively. 

276. The curves represented by the equation 
y n =paf n , 
are called parabolas. This equation may be placed under 
the form 



y=p n x' 
or by placing p n —p\ and —=n', we have 



m 



y=p' x n/ ; 



246 ELEMENTS OF THE 

or finally, by omitting the accents, the form becomes 

y=px n . 
By differentiating, we have 

dy — npx n ~ 1 dx } 

and by substituting this value of dy in the differential of 

the arc, we have 

i 
z=f(l+n 2 p 2 x 2n - 2 fdx. 

The integral of this expression will be expressed in a 

finite number of algebraic terms when is a whole 

number and positive (Art. 235). If we designate such 
whole and positive number by t, we have for the condition 
of an exact integral in algebraic terms, 

1 . 2i+l 



= i, or n — 



2/1-2 ' 2i 7 

and substituting for n, we have 

2t+l. 

y=px 2i ' or y 2i =p* l x* l+l , 

which expresses the relation between x and y when the 
length of the arc can be found in finite algebraic terms. 
There is yet another case in which the integral will be ex- 
pressed in finite and algebraic terms, viz. when o~^~o 

4/71 — ti/ £ 

is a positive whole number (Art. 236 and 235.) 

3 

277. If we make i = 1, we have n = — , and 

2 

y 2 =p 2 x i , 
which is the equation of the cubic parabola. 



INTEGRAL CALCULUS. 247 

Under this supposition, the arc becomes (Art. 217) 

z=f(l + #f4—fl* = 2^20+ TF*f ¥ ; 

and hence, the cubic parabola is rectifiable (Art. 273). 

If we estimate the arc from the vertex of the curve, we 
have x = 0, for z — : hence 

= -iL+C, or C=- 



27 p 2 ' 27p 2 ' 

and consequently, 

8 r/ 9 _ \I "I 

278. If the origin of co-ordinates is at the centre of the 
circle, the equation of the circumference is 

and the value of the arc, 

t==R f*^. 

If the origin be placed on the curve 
y 2 = 2Rx-x 2 , 
dx 



and 



,nf 



V2Rx-, 



both of which expressions may be integrated by series, 
and the length of the arc found approximatively. 

279. It remains to rectify the transcendental curves. 
The differential equation of the cycloid is (Art. 182) 

y d y 



dx 



V 2ry — y 2 



248 ELEMENTS OF THE 

which gives 

da>= y2dy \. 
2ry-y 2 

Substituting this value of da? in the differential of the 
arc, we obtain 



dz = 



= \/dy 2 



+ 



yW 



2 ry — if 



d ys f 



2ry 



2ry-y 



= ^ V^r" = (2r) 7 (2r - y)~dy. 
But (Art. 217) 

f(2r-yyhy=-2(2r-yf+C; 

and hence, 

\_ 

z= -(2r) 2 2^2r -y+ C= - 2 <y/2r(2r - y) + C. 

If now, we estimate 
the arc z from B, the 
point at which y = 2 r, 
we shall have, for z = 0, 
y = 2 r ; hence 

= + C, or C = 0, 

and consequently, the true integral will be 




z— —2^2r{2r — y)\ 

the second member being negative, since the arc is a 
decreasing function of the ordinate y (Art. 31). 

If now, we suppose y to decrease until it becomes 
equal to any ordinate, as DF = ME, DB will be repre- 
sented by z, or by 2 -y/2r(2r — y), and BE = 2r — y. 

But ~BG = BM x BE : hence 

BG = V2r(2r-y\ 



INTEGRAL CALCULUS. 249 

and consequently 

BD = 2BG; 

or the arc of the cycloid, estimated from the vertex of the 
. is equal to twice the corresponding chord of the 
l'< ,ir rating circle : hence, the arc BDA is equal to twice 
the diameter BM ; and the curve ADBL is equal to four 
times the diameter of the generating circle. 

280. The differential of the arc of a spiral, referred to 
polar co-ordinates, is (Art. 202) 



dz — Vdu 2 + u 2 dt 2 . 
Taking the general equation of the spirals 
u — at", 

we have du 2 = nW'W ; 

and substituting for du 2 and u 2 their values, we obtain 



dz = af- l dt-y/ri z + t 2 . 

If we make n = 1, we have the spiral of Archimedes, 
(Art. 191), and the equation becomes 



dz = adt Vl + t 2 ; 

which is of the same form as that of the arc of the com- 
mon parabola (Art. 275). 

281. In the logarithmic spiral, we have t = \ogu, and 
the differential of the arc becomes 

dz = du V2 + C ; 

and if we estimate the arc from the pole, 

z = u-y/2. 



250 ELEMENTS OF THE 

Consequently, the length of the arc estimated from the 
pole to any point of the curve, is equal to the diagonal of 
a square described on the radius-vector, although the 
number of revolutions of the radius-vector between these 
two points is infinite. 

Of the Quadrature of Curves. 

282. The quadrature of a curve is the expression of its 
area. When this expression can be found in finite alge- 
braic terms, the curve is said to be quadrable, and may be 
represented by an equivalent square. 

283. If s represents the area of the segment of a curve, 
and x and y the co-ordinates of any point, we have seen 
(Art. 130), that 

ds = ydx. 

To apply this formula to a given curve : 

1st. Find from the equation of the curve the value of y 
in terms of x, or the value of dx in terms of y, which 
values will be expressed under the forms 

y = f(x), or dx=f(y)dy. 

2d. Substitute the value of y, or the value of dx, in the 
differential of the area : we shall have 

ds = f (x) dx, or ds = f (y) dy : 

the integral of the first form will give the area of the 
curve in terms of the abscissa, and the integral of the 
second will give the area in terms of the ordinate. 



INTEGRAL CALCULUS 251 

284. Let us take, as a first example, the family of para- 
bolas of which the equation is 

y n — px m : 

we shall then have 

i. m 
y=p n x\ 
and 

- - nv n n+n n 

fF(x)dx=zfp n x n dx=— J -—x n = xy+C: 

J v ; JJ/ m + n m + n y 

1 m 

by substituting y for its value, p n x n . 

If, instead of substituting the value of y in the differential 
of the area 

ydx, 

we find the value of dx from the equation 

yn _ p X m^ 



we have 



and consequently, 



dx = 1 — d y. 

m - y 



J y m J ~ ^ m + n - m + n ^ 

p m p m 

n 

by substituting x for its value, — , which is the same re- 

p- 

suit as before found. 

Hence, the area of any portion of a parabola is equal 
to the rectangle described on the abscissa and ordinate 



252 ELEMENTS OF THE 

71 

multiplied by the ratio . The parabolas are there- 

J m-\-n 

fore quadrable. 

In the common parabola, n = 2, m = 1 , and we 
have 

ff(x)dx = -xy, 

that is, the area of a segment is equal to two thirds of 
the area of the rectangle described on the abscissa arid 
ordinate. 

285. If, in the equation 

y n =paT, 

we make n = 1, and m — 1, it will represent a straight 
line passing through the origin of co-ordinates, and we 
shall have 

f f(x)dx-—xy, 

which proves that the area of a triangle is equal to half 
the product of the base and perpendicular. 

286. It is frequently necessary to find the integral or 
function, between certain limits of the variable on which 
it depends. 

A particular notation has been adopted to express such 
integrals. 

Resuming the equation of the common parabola 

y 2 = 2px, 

and substituting in the equation ydx the value of dx •= ?—¥-, 
we have 



P a P 



INTEGRAL CALCULUS. 

or, if the area be estimated from the 
vertex A, we have C = 0, and 

fi/dx = — — . 
JJ 3p 



253 



M 



If now, we wish the area to terminate a 

at any ordinate PM = b, we shall then 

take the integral between the limits of y — and y = b ; 

and, to express that in the differential equation, we write 



l rb ,, V 



3p 

which is read, integral of y 2 dy between the limits y — 
and y — b. 

If we wish the area between the ordinates MP = b, 
MP' = c, we must integrate between the limits y — b, 
y = c. We first integrate between and each limit, viz. : 



AMM'P' = —f C ifdy = 

we then have 

PMM'P = AMMP' - AMP 



Sp 

jL 

3p 



\fb* dy 



V 

c 3 tf 1 
"" 3p ~ Jp ~~3p ^ ~ '' 

287. Let us now determine the area of any portion of 
the space included between the asymptotes and curve of 
an hyperbola. 



254 



ELEMENTS OF THE 



The equation of the hyperbola referrea to its asymp- 
totes (An. Geom. Bk. VI, Prop. IX,) is 

xy = M. 

In the differential of the area of a curve ydx, x and y 
are estimated in parallels to co-ordinate axes, at right an- 
gles to each other. 

The differential of the 
area BCMP, referred to 
the oblique axes AX, 
A Y, is the parallelogram 
PMM'P, of which 
PM=y and PP' = dx. 

If we designate the 
angle YAX=MPX by 
iS, we shall have 

area PMM'P — ydxsmp ; 

M 

and substituting for y its value — , and representing 

x 

the area BCMP by s, we have 
ds = Ms'mB — , 




and 



Cdx 
Msin/3 / — = MsinjSloga? + C. 
J x 



If AC is the semi-transverse axis of the hyperbola, and we 
make AB=l, and estimate the area s from BC, we shall 
have, for x = 1, s = 0, and consequently C = ; and the 
true integral will be 



s = Ms'u\&\ogx. 



INTEGRAL CALCULUS. 255 

But, since ABCD is a rhombus, and M= AB x BC (An. 
Geom. Bk. VI, Prop. IX, Sch. 2), and since AB — 1, we 
have M= 1, and consequently, 

s = sin d \ogx. 

Now, since s, which represents the space BCMP for any 
abscissa x, is equal to the Naperian logarithm of x multi- 
plied by the constant sin/3, s may be regarded as the loga- 
rithm of x taken in a system of which sin£ is the modu- 
lus (Alg. Art. 268). Therefore, any hyperbolic space 
BCMP is the logarithm of the corresponding abscissa 
AP, taken in the system whose modulus is the sine of the 
angle included, between the asymptotes. 

If we would make the spaces the Naperian logarithms 
of the corresponding abscissas, we make sin/3 = 1, which 
corresponds to the equilateral hyperbola. If we would 
make the spaces the common logarithms of the abscissas, 
make sin/3 = 0.43429945, (Alg. Art. 272). 

288. The equation of the circle, when the origin of co- 
ordinates is placed on the circumference, is 



y 2 — 2rx — x 2 , or y = V 2rx — x 2 , 
and hence, the differential of the area is 

dx Vlfrx — x 2 ; 

and this will become, by making x — r — u, 

i 

-fdu^-tfy. 



If we integrate this expression by formula (B, Art. 239, 

17 



256 ELEMENTS OF THE 

we have 
-Jdu{r 2 -u 2 f=-\u{r 2 -u 2 f-]-^fdv{r' u 1 )"^ 



= u V? 2 — u 2 A r 2 I ■ . 

2 2 J Vr*-u 2 



But we have (Art. 253) 

y* — du _ _,/«\ 

and placing for u its value 



fdxV2rx — x 2 = 

(r — x) V 2rx — x 2 -\ r^cos -1 ! ) + C ; 

2 v ; 2 V r J ' 

and taking this integral between the limits x = and 
oc = 2r, we shall have the area of a semicircle. 

For x = 0, the area which is expressed in the first 
member becomes 0, the first term in the second member 
becomes 0, and the second term also becomes 0, since 
the arc whose cosine is 1, is 0: hence the constant 
C = 0. 

If we now make x = 2r, the term 

— (r — x)^/ 2rx — x 2 
2 

reduces to 0, and the second term to 

— r 3 cos- 1 (-l) = — r 2 * (Trig. Art. XIV), 
and consequently, the entire area is equal to r 2 ^, which 



INTEGRAL CALCULUS. 257 

corresponds with a known result (Geom. Bk. V, Prop. XII, 

Coi. 2). 

The equation of the ellipse, the origin of co-ordi- 
nates being at the vertex of the transverse axis (An. Geom. 
Bk. IV, Prop. I. Sch. 8), gives 

B 



A. 

and consequently, the area of the semi-ellipse will be 
equal to 

fydx = — I dxV2Ax— x 2 . 

Integrating, as in the last example, between the limits 
x — 0, and x — 2 A, and multiplying by 2, we find AB* 
for the entire area. This corresponds with a known result 
(An. Geom. Bk. IV, Prop. XIII). 

289. The differential equation of the cycloid (Art. 183) is 

'—" , ydy . 

■yjlry — y 2 
whence 

y 2 dy 



fydx = C- 



^2ry — y 2 

and applying formula E, (Art. 243) twice, it will reduce to 
dy 



i 



and (Art. 226) 



'y-y 
C dy . jy\ 

f , = = ver-sm I — ). 

J V 2ry - y 2 K ^ J 

But we may determine the area of the cycloid in a more 
simple manner by introducing the exterior segment AFKH, 



258 



ELEMENTS OF THE 



Regarding FB as a 
line of abscissas, and de- 
signating any ordinate as 
KII, by z — 2r — y, we 
shall have 




d{AFKH) = zdx. 



But 



whence 



zdx — 



(%r-y)ydy 
V2ry — y 2 



dy^/2ry 



?, 



AFKH = fdyV try— y 2 + C. 

But this integral expresses the area of the segment of a 
circle, of which the abscissa is y and radius r (Art. 288): 
that is, of the segment MIGE. If now, we estimate the 
area of the segment from M, where y — 0, and the area 
AFKH from AF, in which case the area AFKH=0 for 
y = 0, we shall have 

AFKH = MIGE; 

and taking the integral between the limits y = and 
y = 2r, we have 

AFB = semicircle MIGB, 
and consequently, 

area AHBM = A FBM - MIGB. 

But the base of the rectangle A FBM is equal to the semi- 
sircumference of the generating circle, and the altitude is 
equal to the diameter, hence its area is equal to four times 
the area of the semicircle MIGB ; therefore, 



area A HBM = 3 MIGB, 



INTEGRAL CALCULUS. 259 

and consequently, the area AHBL is equal to three times 
the area of the, generating circle. 

290. It now remains to determine the area of the spirals. 
If we represent by s the area described by the radius-vec- 
tor, we have (Art. 203) 

7 u 2 dt 
as = ; 

2 ' 

and placing for u its value at n (Art. 1 89) 

- a 2 t 2n dt , a 2 f n+l 

ds = and 5 == h C, 

2 4w + 2 

and if n is positive C = 0, since the area is when t = 0. 
After one revolution of the radius-vector, t = 2^, and we 
have 

_ a 2 (2*-) 2 " +1 
S ~ 4/1 + 2 ' 

which is the area included within the first spire. 
291. In the spiral of Archimedes (Art. 192) 

a = — and n = 1 ; 
2*r ' 

hence, for this spiral we have 

t 3 
s = 



which becomes -7-, after one revolution of the radius- 
vector ; the unit of the number — being a square whose. 

o 

side is unity. Hence, the area included by the first spire, 
is equal to one third the area of the circle whose radius is 
equal to the radius-vector after the first revolution. 

In the second revolution, the radius-vector describes a 



260 ELEMENTS OF THE 

second time the area described in the first revolution ; and 
in any revolution, it will pass over, or redescribe, all the 
area before generated. Hence, to find the area at the end 
of the mth revolution, we must integrate between the limits 

tz=z{m— 1)2tt and t = m.27r, 

which gives 



m 3 — (m— 1) 



ST. 



3 

If it be required to find the area between any two spires, 
as between the mth and the (m + 1 )th, we have for the 
whole area to the (m + 1 )th spire equal to 

(m + 1 ) 3 — m 3 

8 " ; 

and subtracting the area to the mth spire, gives 
( m+l) 3 -2m 3 + (m- l) 3 

for the area between the mth and (m -f 1 )th spires. 

If we make m=l, we shall have the area between the 
first and second spires equal to 2 c*-: hence, the area be- 
tween the mth and (m + \)th spires, is equal to m times 
the area between the first and second. 

292. In the hyperbolic spiral n = — 1, and we have 

a 2 t~ 2 a 2 

ds = dt and s= . 

2 2t 

The area s will be infinite when t = 0, but we can find 

the area included between any two radius-vectors b and c 

by integrating between the limits t = b, t = c, which will 

give 

a 2 / 1 1 \ 

^-2(T--c-)- 



INTEGRAL CALCULUS. 



261 



du 
293. In the logarithmic spiral t = \ogu : hence, dt = — , 

u 

u 2 dt udu 



hence, 



2 2 ' 

/udu _u 2 n 



and by considering the area s = when u = 0, we have 
C = and 



Determination of the Area of Surfaces of 
Revolution. 



294. If any curve 5MM 7 , be re- 
volved about an axis AX, it will de- 
scribe a surface of revolution, and 
every plane passing through the axis 
AX will intersect the surface in a me- 
ridian curve. It is required to find the 
differential of this surface. For this 



/ 



M 



M 



P P' X 



purpose, make AP = x, PM = y, and PP f = h : we shall 
then have 

PM =/(*) = y, 



262 



ELEMENTS OF THE 



In the revolution of the curve BMM', 
the extremities M and M of the ordi- 
nates MP, M'P', will describe the cir- 
cumferences of two circles, and the 
chord MM' will describe the curved 
surface of the frustum of a cone. The 
surface of this frustum is equal to 
(Geom : Bk. VIII, Prop. IV.) 



{circ. MP + circ. MP') . , n/rnjrf 

x chord MM : 

2 



W 



M 



P P' X 



that is, to 



{2irMP+2*M'P / ) 



X chord MM' =t {MP +M'P>) X chord MM*; 



and by substituting for MP, MP' their values, the expres- 
sion for the area becomes 



dy d?y h 



( 2 ^ + £ /i+ ^r^ +&c cW 



MM 



If now we pass to the limit, by making h = 0, the chord 
MM' will become equal to the arc MM' (Art. 128), and the 
surface of the frustum of the cone will coincide with that 
of the surface described by the curve at the point M. If we 
represent the surface by s and the arc of the curve by z, 
we have, after passing to the limit, 

ds = 27rydz, 

and by substituting for dz its value (Art. 128), we have 



2 . 



ds = 2*y ydx 2, + dy 

whence, the differential of a surface of revolution is equal 
to the circumference of a circle perpendicular to the axis, 
into the differential of the arc of the meridian curve. 



INTEGRAL CALCULUS. 263 

Remark. It should be observed that X is the axis about 
which the curve is revolved. If it were revolved about 
the axis Y, it would be necessary to change x into y and 
y into x. 

295. If a right angled triangle CAB be revolved about 

GO Q 

the perpendicular CA, the hypothenuse CB will describe 
the surface of a right cone. If we represent the base BA 
of the triangle by b. the altitude CA by h, and suppose 
the origin of co-ordinates at the vertex of the angle C, we 



shall have 








x : y : 


: h : b: hence 




b 
y = -x 


and dy — —*dx. 
J h 



Substituting these values of y and dy, in the general for- 
mula, we have 

/2^V^M^W2^^ 

and integrating between the limits x — and x = h, we 

obtain 

nji 

surface of the cone = ^b-y/h 2 -f b 2 = 2?r6 x 

. jy CB 

— circ.AB x . 

2 

296. If a rectangle ABCD be revolved around the side 
AD, we can readily find the surface of the right cylinder 
which will be described by the side BC. 

Let us suppose the axis AD = h, and AB — b : the 
equation of the line DC will be y = b : hence, dy = 0. 
Substituting these values in the general expression of the 
differential of the surface, we have 



J27ry Vdx 2 + dy 2 = f2*bdx = 2*bx A- C ; 



264 ELEMENTS OF THE 

and taking the integral between the limits x = 0,. x — h, 
we have 

surface — 2^bh — circ.AB x AD. 

297. To find the surface of a sphere, let us take the 
equation of the meridian curve, referred to the centre as 
an origin : it is 

a? + y 2 = R 2 , 

and by differentiating, we have 

xdx + ydy = ; 
hence 

ay= and dy i = — - — . 

y y 2 

Substituting for dy its value, in the differential of the 
surface 



ds = 2ny Vdx 2 + dy 2 , 
we obtain 

s=f2iry \Zdx 2 + °^dx 2 =f2irRdx =2ttRx+C. 

If we estimate the surface from the plane passing through 
the centre, and perpendicular to the axis of X, we shall 
have 

5 = for x = 0, and consequently C = 0. 

Now, to find the entire surface of the sphere, we must 
integrate between the limits x = + R and x = — R, and 
then take the sum of the integrals without reference to 
their algebraic signs, for these signs only indicate the po- 
sition of the parts of the surface with respect to the plane 
passing through the centre of the sphere. 



INTEGRAL CALCULUS. 265 

Integrating between the limits 

x — and x = + R> 
we find 

s = 2*R 2 ; 

and integrating between the limits x = and x = — R, 
there results 

s=-2*R 2 ; 
hence, 

surface = 4*rR 2 = 2*R X 2R ; 

that is, equal to four great circles, or equal to the curved 
surface of the circumscribing cylinder. 

298. The two equal integrals 

s = 2nR 2 and s= -2*R 2 

indicate that the surface is symmetrical with respect to the 
plane passing through the centre. 

299. To find the surface of the paraboloid of revolution, 
take the equation of the meridian curve 

y 2 = 2px, 

which being differentiated, gives 



yjk n „A w_iW 



dx = — and dx 2 _ 

P P 



Substituting this value of dx in the differential of the sur- 
face, it reduces to 



*«yV(£pr) d ? = jy d y Vf+f- 



266 ELEMENTS OF THE 

But we have found (Art. 217) 

fjydyV¥+7= f p (f+P 2 f+C: 
hence, 

and if we estimate the surface from the vertex at which 
point y — 0, we shall have, 

= -^-+C, whence, C= - -^-, 

and integrating between the limits 

y=0, y = 6, 



we have 



*=§t(fi*+/) '-^] 



300. To find the surface of an ellipsoid described by 
revolving an ellipse about the transverse axis. 
The equation of the meridian curve is 

whence 

7 B 2 xdx B xdx 



A " y a Va 2 -*? 

substituting the square of this value in the differential of 
the surface and for y its value 
B 



4 VA'-x 2 



we have 

B 



ds=2 rr— dx y/A* - (A 2 - B 2 )^ 



INTEGRAL CALCULUS. 267 



and , = 2. ! sfJF=wfa* s/^ w - * \ 

and if we represent the part without the sign of the inte- 
gral by D, and make 

A 4 



A 2 - B 2 
we shall have 



= R\ 



s = Dfdx<y/K> — 



But the integral of dx^/R 2 — ^ is a circular segment 
of which the abscissa is x, the radius of the circle being 
R. If, then, we estimate the surface of the ellipsoid from 
the plane passing through the centre, and also estimate the 
area of the circular segment from the same point, any 
portion of the surface of the ellipsoid will be equal to the 
corresponding portion of the circle multiplied by the con- 
stant D. Hence, if we integrate the expression 



s=fdx^/R 2 -x 2 

between the limits x — and x = A, and designate 
by D' the corresponding portion of the circle whose 
radius is R, we shall have 

— surface ellipsoid = D x ZK ; 

hence, surface ellipsoid = 2DxD'. 

301. To find the surface described by the revolution of 
the cycloid about its base. 

The differential equation of the cycloid is 

d,= . y dy 

V 2ry — y 2 



268 ELEMENTS OF THE 

Substituting this value of dx in the differential equation 
of the surface, it becomes 

ds _ 2*y2ry T <fy 
V%ry — y 2 

Applying formula (E), Art. 243, we have 



s = 2ny / ~2r 

But, 

y 2 dy r dy 



r"2T/o s- 1 4 r y*^ "1 



J V2>y-f J V2r-y J yK y! K y; ' 



hence, 

5 = 



to JL Q 1— l 

--y\/2fy~^y 2 -- 7 ~r(2r-yY + C. 



If we estimate the surface from the plane passing through 
the centre, we have C = 0, since at this point 5 = 
and y = 2r. If we then integrate between the limits 
y = 2r and y = 0, we have 

6 = — surface = ^r 2 ; hence, 

2 3 

5 = surface = n r 2 , 

3 

that is, the surface described by the cycloid, when it is 
revolved around the base, is equal to 64 thirds of the 
generating circle. 

The minus sign should appear before the integral, since 
the surface is a decreasing function of the variable y 
(Art. 31). 



INTEGRAL CALCULUS. 



269 



Of the Cubature of Solids of Revolution. 

302. The cubature of a solid is the expression of its 
volume or content. 

303. Let u represent the volume or 
solidity generated by the area ABMP, 
when revolved around the axis AX. If 
we make AP = x, PP' — h, we have 
M'P' = F(x + h). Now, the solid gene- 
rated by the area ABMiWP', will ex- 
ceed the solid described by ABMP, by 
the solid described by the area PMM'P' . 

The solid described by the area ABMP is a function of 
x, and the solid described by the area ABMM'P' is a simi- 
lar function of (x + h). If we designate this last by u' f 
we have 




u' =u-\-—h 
ax 



d?u h 2 



+ 



dht h 3 



dx 2 1.2 dx 3 1.2.3 



&c. ; 



hence, the solid described by PMM'P' is 

d 3 u h 3 



du , d?u h 2 
dx dor 1 . 2 



dx 3 1.2.3 



&c. 



Let us now compare the cylinder described by the rectan- 
gle P'M with that described by the rectangle P'C. The 
equation of the curve gives 

MP = y = F{x) M'P'=F(x + h); 
hence, since PP' = h, 

cylinder described by P'M= n[F(x)~fh, 
cylinder described by P'C = n[F(x + h)fh; 



270 ELEMENTS OF THE 

and the ratio of the cylinders is 

[F(x + h)f , 
[_F{x)Y ' 

the limit of which, when h = 0, is unity. 

But the solid described by the area PMM / P / is less 
than one of the cylinders and greater than the other; 
hence, the limit of the ratio, when compared with either 
of them, is unity. Hence, 

du 7 d'hi h 2 s du dhi h B 

n[F(x)] 2 k *[F(x)] 2 

the limit of which, when h = 0, is 



du 

dx 



whence 



and finally 



«[F{x)Y 
du 



1, 



dx = ^(x )f = ^ 



du = iry 2 dx; 



the differential of the solidity -xy 2 dx being a cylinder whose 
base is ny 2 and altitude dx. 

304. Remark. The differential of a solid, generated by 
revolving a curve around the axis of Y, is 

irx 2 dy. 

305. Let it be required to find the solidity of a right 
cylinder with a circular base, the radius of the base being 



INTEGRAL CALCULUS. 271 

r and the altitude h. We have for the differential of the 
solidity 

vry 2 dx, 

and since y = r, it becomes 

and taking the integral between the limits x = and x = h, 
we have 

which expresses the solidity. 

306. To find the solidity of a right cone with a circular 
base, let us represent the altitude by h and the radius of 
the base by r, and let us also suppose the origin of co-or- 
dinates at the vertex. We shall then have 

y = T x and y2 = ~tf^ 

and substituting, the differential of the solidity becomes 

r—KaPdx, 
hr 

and by taking the integral between the limits x = and 
x = h, we obtain 

— -r i ^h = 7zr 2 x-— ; 
3 3 

that is, the area of the base into one third of the altitude. 

307. Let it be required to find the solidity of a prolate 
spheroid, (An : Geom : Bk. IX, Art. 33). 

The equation of a meridian section is 

A 2 y 2 + B 2 x* = A 2 B 2 , 
18 



272 ELEMENTS OF THE 

which gives 

hence the differential of the solidity is 

B 2 

du = 7r-j2(A 2 — a?)dx, 

and by integrating 

If we estimate the solidity from the plane passing through 
the centre, we have for x = 0, u = 0, and consequently 
C = ; and taking the integral between the limits x = 
and x = A, we have 



1 2 

■£- solidity = ~^ 2 x A ; 
4> o 



and consequently 



2 

solidity = y d? 2 x 2A. 



But 7ri? 2 expresses the area of a circle described on the 
conjugate axis, and 2A is the transverse axis : hence, 
the solidity is equal to two-thirds of the circumscribing 
cylinder. 

308. If an ellipse be revolved around the conjugate axis, 
it will describe an oblate spheroid, and the differential of 
the solidity would be 

du = ntfdy : 



INTEGRAL CALCULUS. 273 

and substituting for x 2 , and integrating, we should find 

2 

solidity = — nA 2 x 2B : 

o 

that is, two-thirds of the circumscribing cylinder. 

309. If we compare the two solids together, we find 

oblate spheroid : prolate spheroid : : A : B. 

310. If we make A = B, we obtain the solidity of the 
sphere, which is equal to two-thirds of the circumscribing 
cylinder, or equal to 

311. Let it be required to find the solidity of a para- 
boloid. The equation of a meridian section is 

y 2 = 2px y 

and hence the differential of the solidity is 

du = 2 irpxdx ; hence 

u = npx 2 + C ; 

and estimating the solidity from the vertex, and taking the 
integral between the limits x = and x = h, and designa- 
ting by b the ordinate corresponding to the abscissa x = h, 
we have 

u = nph 2 = vb 2 x — ; 

that is, equal to half the cylinder having an equal base 
and altitude. 

312. Let it be required, as a last example, to determine 



274 



ELEMENTS OF THE 



the solidity of the solid generated by the revolution of the 
cycloid about its base. 

The differential equation of the cycloid is 

dx= , yiy ; 
\2ry — y 2 



hence we have 



du 



Ky' 3 dy 



V2ry — 



ry-y 



which, being integrated by formula (E) Art. 243, and then 
by Art. 226, we find the solidity equal to five-eighths of 
the circumscribing cylinder. 



Of Double Integrals. 

313. Let us, in the first place, consider a solid limited 
by the three co-ordinate planes, and by a curved surface 
which is intersected by the co-ordinate planes in the curves 
CB, BD, DC. 

Through any point of 
the surface, as M, pass 
two planes HQF and 
EPG respectively paral- 
lel to the co-ordinate planes 
ZXy YZ y and intersect- 
ing the surface in the 
curves HMF and EMG. 
The co-ordinates of the 
point M are 

AP=x, PM=y, MM'=z./Q 




INTEGRAL CALCULUS. 275 

It is now evident that the solid whose base on the co-ordi- 
nate plane YX is the rectangle AQM'P, may be extended 
indefinitely in the direction of the axis of X without chang- 
ing the value of y, or indefinitely in the direction of Y 
without changing x. Hence, x and y may be regarded 
as independent variables. 

If, for example, we suppose y to remain constant, and x 
to receive an increment Pp — h, the solid whose base is 
the rectangle AQM'P, will be increased by the solid 
whose base is the rectangle M'm'pP ; and if we suppose 
x to remain constant, and y to receive an increment 
Qq — k, the first solid will be increased by the solid whose 
base is the rectangle Qqn'M'. 

But if we suppose x and y to receive their increments 
at the same time, the new solid will still be bounded by 
the parallel planes epg, hqf, and will differ from the prim- 
itive solid not only by the two solids before named, but 
also by the solid whose base is the rectangle n'M'mfN'. 
This last solid is the increment of the solid whose base is 
the rectangle M'Ppm', when we suppose y to vary ; or 
the increment of the solid whose base is the rectangle 
Qqn f M f , when we suppose x to vary. 

Let us represent by u the solid whose base is the rect- 
angle A QM'P ; u will then be a function of x and y, and 
the difference between the values of the increments of u, 
under the supposition that x and y vary separately ; and 
under the supposition that they vary together, will be equal 
to the solid whose base is the rectangle n'M'm'N'. By 
taking this difference (Art. 83) we have 

solidn'JV»m'J^...Jtf=4^Aifc+i^A%+i-^^+&c. : 
dxdij 2 dxrdij 2 dxdif 



276 ELEMENTS OF THE 

hence, 

solid n'N , m f M'...M d?u 1 <Pu 1 d?u 

hk " dxdy 2 dx*dy 2 cMz/ 2 

and passing to the limit, by making h = and k=0, the 

second member becomes -= — r-. 

dxdy 

As regards the first member, the rectangle 

n'N'm'M* = hxk, 

and the altitude of the solid becomes equal to M / M = z 
when we pass to the limit : hence 

d?u 
dxdy 

314. Although the differential coefficient 

= z. 



dxdy 

has been determined by regarding u as a function of two 
variables, we can nevertheless return to the function u by 
the methods which have been explained for integrating a 
function of a single variable. 
For we have 

d?u \dx) 

-z; 



dxdy dy 

hence 

and integrating under the supposition that x remains con- 



INTEGRAL CALCULUS. 277 

stant, and y varies, we have 

whence 

— dx = dxfzdy + X! dx ; 

CLX 

and if we integrate this last expression under the supposi- 
tion of x being the variable, and make J ' X! dx = X, 

u = fdxfzdy + X + Y. 

It is plain that the constant, which is added to complete 
the first integral, may contain x in any manner whatever; 
and that which is added in the second integral, may contain 
y : the first ivill disappear when we differentiate with 
respect to y, and the second when we differentiate with 
respect to x. 

The order of integration is not material. If we first 
integrate with respect to x, we can write 

d?u _ \dy) . 
dxdy dx 

and by integrating, we find 

—-=fzdx, u= fdyfzdx : 

hence we may write 

u = ffzdy dx, or u = ffzdxdy, 

which indicates that there are two integrations to be per- 
formed, one with respect to x, and the other with respect 
to y. 



278 ELEMENTS OF THE 

315. If we consider the differentials as the indefinitely 
small increments of the variables on which they depend, 
we may regard the prism whose base is the rectangle 
n f N'm f M f , as composed of an indefinite number of small 
prisms, having equal bases, and a common altitude dz. 
Each one of these prisms will be expressed by dxdydz, 
and we shall obtain their sum by integrating with respect 
to z between the limits z = and z = MM f , which 
will give 

J dxdydz = zdxdy. 

316. It is plain that zdx is the differential of the area 
of the section made by the plane HQF parallel to the 
co-ordinate plane ZX ; and consequently 

/zc?a? = area of the section HQF. 

Hence, (Jzdx)dy is equal to the elementary solid in- 
cluded between the parallel planes HQF, hqf, or 

f(fzdx)dy =ff zdxdy 

is equal to the solid which is limited by the surface and 
the three co-ordinate planes. If we consider a section 
of the solid parallel to the co-ordinate plane YZ, we have 
fzdy = area of the section EPG, and ff zdxdy = solidity 
of the solid. 

317. Let us suppose, as a first example, that 

_ 1 

we shall then have 



INTEGRAL CALCULUS. 279 

Let us now integrate under the supposition that x is con- 
stant ; we then have 

J xr + y x x 

in which X! represents an arbitrary function of x. If we 
now make fX'dx = X, and integrate again under the 
supposition that a? is a variable, we have 

J X X 

dx \i 

The integral of — tang" 1 — is obtained in a series by 
x x 

substituting the value of (Art. 228), 

° x x Sx 6 bxr 7x 7 

and since, in integrating with respect to x, we must add 
an arbitrary function of y, which we will represent by Y, 
we shall obtain 

r rdxdy . y . f £_>_£_ _ &c 

J J a? + y*- ^ a? 9* 3 25* 5 + 49a? 7 

We shall obtain the same result by integrating in the in- 
verse order, viz., by first supposing y to be constant. 
Under this supposition 

J ar + y z y y 



280 ELEMENTS OF THE 

then integrating with respect to x, 



f 



% tang -iiL+y. 

y y 



But by observing that (Trig. Art. XVIII), 

tang ' — = tang — , 

we shall have, after the second integration, and the addi- 
tion of an arbitrary function of x, 

and as we can include the term — logy in the arbitrary 
function F, this result may be placed under the form 

which is the same as the result before obtained, as may be 
shown by placing for tang" 1 — its value, multiplying each 

u d y J • x 

term by — , and integrating. 

318. When we consider 

ffzdxdy 

as expressing the solidity of a solid, it is necessary to con- 
sider the limits between which each integral is taken, and 
these limits will depend on the nature of the solid whose 
cubature is to be determined. Let it bt iequired, for ex- 



INTEGRAL CALCULUS. 281 

ample, to find the solidity of a sphere, of which the centre 
is at the origin of co-ordinates. Designating the radius 
by R, we have 

a* + tf + s* = R\ 

and consequently, 

ffzdxdy =ffdxdy Ylt 2 - x 2 - y 2 . 

If now, we suppose y constant, and make R 2 — y 2 = R /2 , 
and then integrate with respect to x, we have 

fdx VR 2 -x 2 - y 2 = fdx VR' 2 - x\ 

and integrating this last expression, first by formula (B) 
Art. 239, and then by Art. 220, we have 

fdxVR' 2 -x 2 = ~VR /2 - x 2 + ^R /2 sm~ l ^ } + Y; 

and substituting for R' 2 its value, we obtain 

/dxVR'-xi-tf^VR'-xt-tf+^R'-tf) sin-i(:^==) +F. 

It should be remarked, that fzdx expresses the area of 
a section of the sphere parallel to the co-ordinate plane 
ZX 7 for any ordinate y = A Q, and to obtain this area we 
must integrate between the limits x = and x = QF. 
But since the point F is in the co-ordinate plane YX, 
we have for this point z = 0, and the equation of the sur- 
face gives 

QF = x=VrF^~i; 

therefore, for every value of y the integral f zdx must be 
taken between the limits x = and x = VR 2 — y 2 . Inte- 



282 ELEMENTS OF THE 

grating between these limits we have 



fdxVR 2 -* 2 — */ 2 = — (R 2 -y 2 )s'm- l (l) 

since, sin -1 (l) = — : 

hence, 
fdyfzdx = ^-fdy(R' - f) = ^(ll*y - V j) + X, 

and taking this last integral between the limits y = and 
y = AC = R, we obtain 



6 ' 
which represents that part of the sphere that is contained 
in the first angle of the co-ordinate planes, or one-eighth 
of the entire solidity. Hence, 

4.1 

solidity of the sphere = — R 3 ^ — — D 3 **. 

We might at once find the solidity of the hemisphere 
which is above the horizontal plane YX, by integrating 
between the limits 



x — — VR 2 — y 2 and x — + VR 2 — y 2 . 
Taking the integral between the limits 

x — and x — — V R 2 — y 2 , 

we have fzdx — — — (R 2 — y 2 ) ; 

and between the limits 



x = and x = -f- V R 2 - y 2 , 



INTEGRAL CALCULUS. 283 



we have Jzdx z=—(R* — y 2 ); 

hence, between the extreme limits, we have 

fzdx = ^(R*-f). 

Then taking the integral 

fdyJzdz = ^fdy{R>-f) 

between the limits 

y = — R and y = + R, 
we find the solidity to be 

or the solidity of the entire sphere is, 



THE END 





































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